9. Relations and functions

# 9. Relations and functions ## Relation between 2 sets and in a set. ### Introduction Welcome to this lesson on relations! In our daily lives, we constantly encounter relationships between different things, people, or ideas. For example, a student is related to their school, a child is related to their parents, or a price is related to a product. In mathematics, we use the concept of 'relations' to formally describe how elements from one set are connected to elements from another set, or how elements within the same set are connected. Understanding relations is fundamental for later topics like functions, which are a special type of relation. ### Key Concepts #### What is a Set? A set is simply a well-defined collection of distinct objects. These objects are called elements or members of the set. For example: * Set A: \( A = \{1, 2, 3, 4\} \) * Set B: \( B = \{\text{apple, banana, cherry}\} \) #### Ordered Pairs An **ordered pair** is a pair of objects \( (a, b) \) where the order of the objects matters. This is different from a set \( \{a, b\} \) where \( \{a, b\} = \{b, a\} \). In an ordered pair, the first element is \( a \) and the second element is \( b \). * Example: \( (1, 2) \) is an ordered pair. Note that \( (1, 2) \neq (2, 1) \). #### Cartesian Product of Two Sets Before we define a relation, we need to understand the **Cartesian product**. The Cartesian product of two non-empty sets, say \( A \) and \( B \), is the set of all possible ordered pairs \( (a, b) \) where \( a \) is an element from set \( A \) and \( b \) is an element from set \( B \). It is denoted by \( A \times B \) (read as "A cross B"). Mathematically, this is written as: \[ A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\} \] If set \( A \) has \( m \) elements and set \( B \) has \( n \) elements, then the Cartesian product \( A \times B \) will have \( mn \) elements. * **Example:** Let \( A = \{1, 2\} \) and \( B = \{x, y, z\} \). Then, the Cartesian product \( A \times B \) is: \( A \times B = \{(1, x), (1, y), (1, z), (2, x), (2, y), (2, z)\} \) Here, \( |A|=2 \) and \( |B|=3 \), so \( |A \times B| = 2 \times 3 = 6 \) elements. #### Definition of a Relation between Two Sets A **relation** \( R \) from a non-empty set \( A \) to a non-empty set \( B \) is any subset of the Cartesian product \( A \times B \). This means that a relation consists of some (or all) of the ordered pairs from \( A \times B \). If \( (a, b) \in R \), we say that \( a \) is related to \( b \), and we can write this as \( a R b \). * **Example:** Let \( A = \{1, 2, 3\} \) and \( B = \{2, 4, 6\} \). Consider the relation \( R \) from \( A \) to \( B \) defined by "\( a \) is a factor of \( b \)". First, find \( A \times B \): \( A \times B = \{(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 2), (3, 4), (3, 6)\} \) Now, select the ordered pairs \( (a, b) \) where \( a \) is a factor of \( b \): \( R = \{(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 6)\} \) Notice that \( R \) is a subset of \( A \times B \). #### Representing Relations Relations can be represented in several ways: 1. **Listing Ordered Pairs:** As shown above, we can simply list all the ordered pairs that belong to the relation. 2. **Arrow Diagram (Sagittal Diagram):** Draw two ovals representing the sets. Draw arrows from elements in the first set to related elements in the second set. 3. **Table:** Create a table with two columns, one for elements of set A and one for elements of set B, showing the connections. 4. **Rule or Description:** Define the relation using a mathematical rule or a descriptive phrase (e.g., "is less than", "is a multiple of"). #### Domain and Range of a Relation For any relation \( R \) from set \( A \) to set \( B \): * The **domain** of \( R \) is the set of all first elements (x-coordinates) of the ordered pairs in \( R \). \( \text{Domain}(R) = \{a \mid (a, b) \in R \text{ for some } b \in B\} \) * The **range** of \( R \) is the set of all second elements (y-coordinates) of the ordered pairs in \( R \). \( \text{Range}(R) = \{b \mid (a, b) \in R \text{ for some } a \in A\} \) * **Example:** For \( R = \{(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 6)\} \) from the previous example: * Domain of \( R = \{1, 2, 3\} \) * Range of \( R = \{2, 4, 6\} \) #### Relation in a Set (Binary Relation on a Set) A relation can also exist **within a single set**. This is called a binary relation on a set \( A \), and it is simply a relation from \( A \) to \( A \). In other words, it is a subset of the Cartesian product \( A \times A \). * **Example:** Let \( A = \{1, 2, 3, 4\} \). Consider the relation \( R \) on \( A \) defined by "\( x \) is less than \( y \)" where \( x, y \in A \). \( R = \{(1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4)\} \) Here, the domain is \( \{1, 2, 3\} \) and the range is \( \{2, 3, 4\} \). ### Vocabulary * **Set:** A collection of distinct objects. * **Element:** An object belonging to a set. * **Ordered Pair:** A pair of objects \( (a, b) \) where the order of elements matters. * **Cartesian Product (\( A \times B \)):** The set of all possible ordered pairs \( (a, b) \) where \( a \in A \) and \( b \in B \). * **Relation (\( R \)):** A subset of the Cartesian product \( A \times B \) (or \( A \times A \)). It defines a specific connection between elements. * **Domain of a Relation:** The set of all first elements of the ordered pairs in the relation. * **Range of a Relation:** The set of all second elements of the ordered pairs in the relation. * **Arrow Diagram (Sagittal Diagram):** A visual representation of a relation using ovals for sets and arrows for connections. ### Examples #### Example 1: Relation between two sets Given set \( P = \{2, 3, 5\} \) and set \( Q = \{4, 6, 9, 10\} \). Let \( R \) be the relation from \( P \) to \( Q \) defined by "\( x \) is a factor of \( y \)". 1. **List the ordered pairs for \( R \):** * For \( x=2 \): 2 is a factor of 4, 6, 10. So \( (2, 4), (2, 6), (2, 10) \) are in \( R \). * For \( x=3 \): 3 is a factor of 6, 9. So \( (3, 6), (3, 9) \) are in \( R \). * For \( x=5 \): 5 is a factor of 10. So \( (5, 10) \) is in \( R \). Thus, \( R = \{(2, 4), (2, 6), (2, 10), (3, 6), (3, 9), (5, 10)\} \). 2. **Draw an arrow diagram:** (Imagine two ovals, one for P and one for Q. Draw arrows from 2 to 4, 6, 10; from 3 to 6, 9; from 5 to 10.) 3. **State the domain and range of \( R \):** * Domain of \( R = \{2, 3, 5\} \) * Range of \( R = \{4, 6, 9, 10\} \) #### Example 2: Relation in a set Given set \( M = \{1, 2, 3, 4\} \). Let \( S \) be the relation on \( M \) defined by "\( x + y = 5 \)" where \( x, y \in M \). 1. **List the ordered pairs for \( S \):** * If \( x=1 \), \( y=4 \). So \( (1, 4) \) is in \( S \). * If \( x=2 \), \( y=3 \). So \( (2, 3) \) is in \( S \). * If \( x=3 \), \( y=2 \). So \( (3, 2) \) is in \( S \). * If \( x=4 \), \( y=1 \). So \( (4, 1) \) is in \( S \). Thus, \( S = \{(1, 4), (2, 3), (3, 2), (4, 1)\} \). 2. **State the domain and range of \( S \):** * Domain of \( S = \{1, 2, 3, 4\} \) * Range of \( S = \{1, 2, 3, 4\} \) ### Exercises 1. Let \( A = \{a, b, c\} \) and \( B = \{1, 2, 3, 4\} \). Let \( R \) be the relation from \( A \) to \( B \) defined as \( R = \{(a, 1), (a, 3), (b, 2), (c, 4)\} \). a. State the domain of \( R \). b. State the range of \( R \). c. Draw an arrow diagram for \( R \). 2. Given the relation \( T = \{(4, 2), (6, 3), (8, 4), (10, 5)\} \). a. Identify the sets \( X \) and \( Y \) such that \( T \) is a relation from \( X \) to \( Y \). b. Describe the rule or description of the relation \( T \) in words. c. What is the domain and range of \( T \)? 3. Let \( K = \{5, 6, 7, 8\} \). Let \( L \) be a relation on \( K \) defined by "\( x \) is greater than \( y \)" where \( x, y \in K \). a. List all the ordered pairs in \( L \). b. State the domain and range of \( L \). ### Assignment 1. Consider sets \( X = \{ \text{Monday, Tuesday, Wednesday} \} \) and \( Y = \{ \text{Lunch, Dinner, Breakfast} \} \). Let \( F \) be the relation from \( X \) to \( Y \) defined by "The meal \( y \) is typically eaten on day \( x \)". Assume standard meal times. a. List the ordered pairs in relation \( F \). b. Draw an arrow diagram for \( F \). c. What is the domain and range of \( F \)? 2. Given the set \( S = \{ \text{red, blue, green, yellow} \} \). Let \( C \) be a relation on \( S \) defined by "\( x \) and \( y \) are primary colours". a. List the ordered pairs in relation \( C \). b. Is this a symmetric relation? (i.e., if \( (x,y) \in C \) then \( (y,x) \in C \) ) c. What is the domain and range of \( C \)? *(Hint: Primary colours are red, blue, yellow.)* ### Summary In this lesson, we explored the concept of relations in mathematics. We learned that: * An **ordered pair** \( (a, b) \) specifies a unique connection where order matters. * The **Cartesian product** \( A \times B \) lists all possible ordered pairs between elements of set \( A \) and set \( B \). * A **relation** \( R \) from set \( A \) to set \( B \) is simply any subset of the Cartesian product \( A \times B \). * Relations can be represented by **listing ordered pairs**, using **arrow diagrams**, or by a **rule/description**. * The **domain** of a relation is the set of all first elements, and the **range** is the set of all second elements of the ordered pairs in the relation. * A **relation in a set** is a relation where both the domain and codomain are the same set (i.e., a subset of \( A \times A \)). Understanding relations is a crucial step towards understanding more complex mathematical structures like functions, which we will explore in future lessons. ## Cartesian product A×B and ### Introduction Welcome to this lesson on the Cartesian product! In mathematics, we often deal with sets of objects. Today, we will learn how to combine elements from two different sets to form new pairs. This concept, called the Cartesian product, is fundamental in various areas of mathematics, including relations, functions, and coordinate geometry. By the end of this lesson, you will be able to define, list, and understand the properties of the Cartesian product of two sets. ### Key Concepts #### What is a Set? A set is a well-defined collection of distinct objects. These objects are called elements or members of the set. * **Examples:** * \( A = \{1, 2, 3\} \) (Set A contains the numbers 1, 2, and 3) * \( B = \{\text{apple, banana, cherry}\} \) (Set B contains three fruits) #### Ordered Pairs An ordered pair is a pair of objects where the order in which the objects appear matters. It is written as \( (a, b) \), where \( a \) is the first element and \( b \) is the second element. * **Key Idea:** \( (a, b) \) is generally not the same as \( (b, a) \) unless \( a = b \). * **Example:** \( (2, 3) \) is different from \( (3, 2) \). #### Definition of Cartesian Product A×B If \( A \) and \( B \) are two non-empty sets, the Cartesian product of \( A \) and \( B \), denoted as \( A \times B \) (read as "A cross B"), is the set of all possible ordered pairs \( (a, b) \) such that \( a \) is an element of \( A \) and \( b \) is an element of \( B \). * **Formula:** \[ A \times B = \{ (a,b) \mid a \in A \text{ and } b \in B \} \] #### Properties of Cartesian Product * **Non-commutative:** In general, \( A \times B \neq B \times A \). The order of the sets matters unless \( A = B \) or one of the sets is empty. * **Cardinality of Cartesian Product:** If set \( A \) has \( m \) elements (denoted as \( |A| = m \)) and set \( B \) has \( n \) elements (denoted as \( |B| = n \)), then the Cartesian product \( A \times B \) will have \( m \times n \) elements. \[ |A \times B| = |A| \times |B| \] * **Empty Set:** If either \( A \) or \( B \) is an empty set (denoted as \( \emptyset \)), then their Cartesian product is also an empty set. \[ A \times \emptyset = \emptyset \quad \text{and} \quad \emptyset \times B = \emptyset \] ### Vocabulary * **Set:** A collection of distinct objects or elements. * **Element:** An individual object belonging to a set. * **Ordered Pair:** A pair of elements \( (a, b) \) where the order of \( a \) and \( b \) is significant. * **Cartesian Product:** The set of all possible ordered pairs formed by taking the first element from one set and the second element from another set. * **Cardinality:** The number of elements in a set, denoted by \( |A| \) for set \( A \). ### Examples **Example 1:** Finding \( A \times B \) and \( B \times A \) Let \( A = \{1, 2\} \) and \( B = \{a, b\} \). * **To find \( A \times B \):** Take each element from \( A \) and pair it with each element from \( B \). \( A \times B = \{ (1,a), (1,b), (2,a), (2,b) \} \) * **To find \( B \times A \):** Take each element from \( B \) and pair it with each element from \( A \). \( B \times A = \{ (a,1), (a,2), (b,1), (b,2) \} \) **Observation:** Notice that \( A \times B \neq B \times A \) because the ordered pairs are different. For instance, \( (1,a) \neq (a,1) \). **Example 2:** Calculating Cardinality Let \( P = \{\text{red, blue, green}\} \) and \( Q = \{\text{car, bike}\} \). * \( |P| = 3 \) (P has 3 elements) * \( |Q| = 2 \) (Q has 2 elements) * **To find \( P \times Q \):** \( P \times Q = \{ (\text{red, car}), (\text{red, bike}), (\text{blue, car}), (\text{blue, bike}), (\text{green, car}), (\text{green, bike}) \} \) * **Cardinality of \( P \times Q \):** \( |P \times Q| = |P| \times |Q| = 3 \times 2 = 6 \) We can see there are indeed 6 ordered pairs. **Example 3:** Real-world application A restaurant offers a choice of 2 main courses (Chicken, Fish) and 3 drink options (Juice, Water, Soda). What are all the possible meal combinations (main course, drink)? Let \( M = \{\text{Chicken, Fish}\} \) and \( D = \{\text{Juice, Water, Soda}\} \). \( M \times D = \{ (\text{Chicken, Juice}), (\text{Chicken, Water}), (\text{Chicken, Soda}), (\text{Fish, Juice}), (\text{Fish, Water}), (\text{Fish, Soda}) \} \) There are \( 2 \times 3 = 6 \) possible meal combinations. ### Exercises 1. Given \( A = \{p, q\} \) and \( B = \{x, y, z\} \): * a) Find \( A \times B \). * b) Find \( B \times A \). * c) Is \( A \times B = B \times A \)? Explain why or why not. 2. Let \( C = \{\text{pen, pencil}\} \) and \( D = \{\text{blue}\} \). List the elements of \( C \times D \). 3. If \( |X| = 4 \) and \( |Y| = 5 \), what is the cardinality of \( X \times Y \)? 4. If \( |P| = 3 \) and \( |Q \times P| = 12 \), what is \( |Q| \)? 5. True or False: If \( S \) is an empty set, then \( S \times T = \emptyset \) for any set \( T \). ### Assignment 1. Let \( S = \{\text{even numbers less than 7}\} \) and \( T = \{\text{vowels}\} \). * a) List the elements of set \( S \) and set \( T \). * b) Find \( S \times T \). * c) Find \( T \times S \). * d) What is \( |S \times T| \)? 2. Given \( A = \{1, 2, 3\} \) and \( B = \{1, 2\} \), find \( A \times B \). * Is \( (3,2) \in A \times B \)? * Is \( (2,3) \in A \times B \)? 3. A shirt company offers 4 different sizes (S, M, L, XL) and 5 different colors (Red, Blue, Green, White, Black). How many different types of shirts (size, color) are possible? List some of the possible types using the concept of Cartesian product. 4. Consider the set \( R = \{a, b, c\} \). Find \( R \times R \). What is \( |R \times R| \)? ### Summary In this lesson, we explored the Cartesian product, a fundamental concept for combining sets. We learned that: * The **Cartesian product** \( A \times B \) is the set of all ordered pairs \( (a, b) \) where \( a \) comes from set \( A \) and \( b \) comes from set \( B \). * **Ordered pairs** are crucial because the order of elements matters, meaning \( (a, b) \neq (b, a) \) unless \( a = b \). * The **cardinality** (number of elements) of \( A \times B \) is the product of the cardinalities of \( A \) and \( B \), i.e., \( |A \times B| = |A| \times |B| \). * The Cartesian product is **not commutative**; \( A \times B \) is generally different from \( B \times A \). * If either set is empty, their Cartesian product is also the empty set. Understanding the Cartesian product is a key step towards studying relations and functions in mathematics. ## Types of relations ### Introduction Welcome to this lesson on **Relations and Functions**! In mathematics, we often look for connections between different quantities or sets of items. These connections are what we call **relations**. Understanding different types of relations helps us to categorize these connections and, importantly, to identify a special type of relation known as a **function**, which is fundamental in many areas of mathematics and science. In this lesson, you will learn to identify and describe various types of relations, differentiating between them based on how elements from one set are linked to elements in another set. ### Key Concepts #### What is a Relation? A **relation** is simply a set of ordered pairs, \((x, y)\), showing a relationship between two sets. The first element of each pair, \(x\), comes from the **domain**, and the second element, \(y\), comes from the **range**. #### Ways to Represent Relations Relations can be represented in several ways: * **Ordered Pairs:** A list of pairs, e.g., \(\{(1, A), (2, B), (3, C)\} * **Arrow Diagrams:** Visual representation where arrows connect elements from the domain to elements in the range. * **Graphs:** Plotting the ordered pairs on a Cartesian coordinate plane. * **Equations:** An algebraic rule that defines the relationship, e.g., \(y = 2x + 1\). #### Types of Relations There are four main types of relations, categorized by how elements of the domain relate to elements of the range: ##### 1. One-to-one Relation * **Definition:** Each element in the domain is related to exactly one unique element in the range. * **Characteristics:** No two ordered pairs have the same first element or the same second element. * **Example:** * \(\{(1, 5), (2, 6), (3, 7)\} * A student and their unique student ID number. ##### 2. One-to-many Relation * **Definition:** One element in the domain is related to more than one element in the range. * **Characteristics:** At least one first element is paired with more than one second element. * **Example:** * \(\{(A, P), (A, Q), (B, R)\} * A father and his children (one father can have many children). * The relation \(x = y^2\) where \(x\) is the domain element and \(y\) is the range element. For \(x=4\), \(y\) can be \(2\) or \(-2\). ##### 3. Many-to-one Relation * **Definition:** More than one element in the domain is related to exactly one element in the range. * **Characteristics:** At least two different first elements are paired with the same second element. * **Example:** * \(\{(1, X), (2, X), (3, Y)\} * Students and their favorite subject (many students can have the same favorite subject, e.g., three students like Mathematics). * The relation \(y = x^2\). For \(y=4\), \(x\) can be \(2\) or \(-2\). (Here, \(x\) is the domain, \(y\) is the range). ##### 4. Many-to-many Relation * **Definition:** More than one element in the domain is related to more than one element in the range. * **Characteristics:** Combines features of one-to-many and many-to-one relations. Multiple first elements are paired with multiple second elements. * **Example:** * \(\{(1, A), (1, B), (2, C), (3, A)\} * Countries and the languages spoken in them (many countries speak many languages, and many languages are spoken in many countries). * Students and the clubs they join (many students can join many clubs, and many clubs have many students). #### Relations vs. Functions A **function** is a special type of relation. For a relation to be a function, each element in the **domain** must be mapped to exactly one element in the **range**. * **Functions:** One-to-one and Many-to-one relations are functions because each input \(x\) has only one output \(y\). * **Not Functions:** One-to-many and Many-to-many relations are *not* functions because at least one input \(x\) has more than one output \(y\). ### Vocabulary * **Relation:** A set of ordered pairs connecting elements from two sets. * **Domain:** The set of all first elements (x-values) in the ordered pairs of a relation. * **Range:** The set of all second elements (y-values) in the ordered pairs of a relation. * **Codomain:** The set of all possible output values. The range is a subset of the codomain. * **Function:** A special type of relation where each element in the domain is mapped to exactly one element in the range. * **Ordered Pair:** A pair of numbers \((x, y)\) where the order matters. ### Examples **Example 1: Identifying Relation Types** Consider the following relations: 1. \(R_1 = \{(2, 4), (3, 6), (4, 8)\} * **Type:** One-to-one. Each domain element \((2, 3, 4)\) maps to a unique range element \((4, 6, 8)\). * **Is it a function?** Yes. 2. \(R_2 = \{(1, A), (1, B), (2, C)\} * **Type:** One-to-many. The domain element \(1\) maps to two different range elements \(A\) and \(B\). * **Is it a function?** No. 3. \(R_3 = \{(P, X), (Q, X), (R, Y)\} * **Type:** Many-to-one. Domain elements \(P\) and \(Q\) both map to the same range element \(X\). * **Is it a function?** Yes. 4. \(R_4 = \{(A, 1), (B, 1), (A, 2)\} * **Type:** Many-to-many. Element \(A\) maps to \(1\) and \(2\); elements \(A\) and \(B\) both map to \(1\). * **Is it a function?** No. **Example 2: Real-world scenarios** 1. **Relation:** Each student in a class to their height. * **Type:** Many-to-one. Several students can have the same height. * **Is it a function?** Yes. 2. **Relation:** The age of a person to the years they were born. * **Type:** One-to-many. A specific age (e.g., 15 years old) could correspond to different birth years depending on the current year. * **Is it a function?** No. ### Exercises Determine the type of relation for each of the following and state whether it is a function or not: 1. \(\{(John, 101), (Mary, 102), (Peter, 103)\} 2. \(\{(Football, 11), (Basketball, 5), (Rugby, 15)\} 3. \(\{(Nairobi, Kenya), (Kampala, Uganda), (Dodoma, Tanzania)\} 4. \(\{(Car, Wheel), (Car, Engine), (Car, Door)\} 5. \(\{(Mother, Child1), (Mother, Child2), (Father, Child1)\} 6. Consider the relation given by the equation \(y = |x|\). 7. Consider the relation given by the equation \(x^2 + y^2 = 9\). ### Assignment 1. For each of the following, draw an arrow diagram and identify the type of relation. State whether it is a function. a. \(\{(A, apple), (B, banana), (C, cherry), (D, apple)\} b. \(\{(10, pen), (10, pencil), (20, book)\} c. The relation between students and the subjects they are taking in school. 2. Provide a real-world example for each of the four types of relations (one-to-one, one-to-many, many-to-one, many-to-many) that was not discussed in the lesson. 3. Explain in your own words why a one-to-many relation cannot be a function, but a many-to-one relation can be a function. ### Summary In this lesson, we explored the concept of **relations** as connections between elements of two sets, represented by ordered pairs. We learned about the four primary types of relations: * **One-to-one:** Each domain element links to a unique range element. * **One-to-many:** A domain element links to multiple range elements. * **Many-to-one:** Multiple domain elements link to a single range element. * **Many-to-many:** Multiple domain elements link to multiple range elements. Crucially, we distinguished between general relations and **functions**. A relation is only a function if every element in the domain is mapped to **exactly one** element in the range. Therefore, one-to-one and many-to-one relations are functions, while one-to-many and many-to-many relations are not functions. Understanding these distinctions is a fundamental step towards more advanced mathematical concepts. ## Properties of relations in a set. ### Introduction Welcome to this lesson on the properties of relations! In mathematics, a relation describes how elements in a set (or between sets) are connected. Just like people can be related (e.g., 'is a brother of', 'is taller than'), numbers and objects can also have relationships. Understanding these properties helps us classify relations and uncover deeper mathematical structures. We will explore three fundamental properties: reflexive, symmetric, and transitive, and see how they combine to form a special type of relation called an equivalence relation. ### Key Concepts #### What is a Relation? A relation R from a set A to a set B is simply a collection of ordered pairs \( (a, b) \) where \( a \) is an element of A and \( b \) is an element of B. This means R is a subset of the Cartesian product \( A \times B \). If a relation R is defined on a single set A, then it is a subset of \( A \times A \). * **Notation:** If an ordered pair \( (a, b) \) is in relation R, we often write \( a R b \) (read as 'a is related to b'). #### Properties of Relations in a Set A ##### Reflexive Relation A relation R on a set A is **reflexive** if every element in A is related to itself. * **Definition:** For every element \( a \) in set A, \( (a, a) \) must be an element of R. Or, \( a R a \) for all \( a otin A \). * **Example:** The relation "is equal to" (\( = \)) on the set of real numbers is reflexive because every real number is equal to itself (e.g., \( 5 = 5 \)). ##### Symmetric Relation A relation R on a set A is **symmetric** if whenever an element \( a \) is related to an element \( b \), then \( b \) is also related to \( a \). * **Definition:** If \( (a, b) \) is an element of R, then \( (b, a) \) must also be an element of R. Or, if \( a R b \), then \( b R a \). * **Example:** The relation "is a sibling of" on a set of people is symmetric. If Person A is a sibling of Person B, then Person B is also a sibling of Person A. * **Non-example:** The relation "is taller than" is not symmetric. If A is taller than B, B cannot be taller than A. ##### Transitive Relation A relation R on a set A is **transitive** if whenever \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) must also be related to \( c \). * **Definition:** If \( (a, b) \) is an element of R and \( (b, c) \) is an element of R, then \( (a, c) \) must also be an element of R. Or, if \( a R b \) and \( b R c \), then \( a R c \). * **Example:** The relation "is less than" (\( < \)) on the set of integers is transitive. If \( a < b \) and \( b < c \), then it follows that \( a < c \). * **Non-example:** The relation "is a friend of" might not be transitive. If A is a friend of B, and B is a friend of C, it doesn't necessarily mean A is a friend of C. ##### Equivalence Relation A relation R on a set A is called an **equivalence relation** if it possesses all three properties: 1. **Reflexive** 2. **Symmetric** 3. **Transitive** * **Example:** The relation "is congruent to" on the set of all triangles is an equivalence relation. * Every triangle is congruent to itself (Reflexive). * If triangle T1 is congruent to T2, then T2 is congruent to T1 (Symmetric). * If T1 is congruent to T2, and T2 is congruent to T3, then T1 is congruent to T3 (Transitive). * **Equivalence Classes:** An equivalence relation partitions a set into disjoint subsets called **equivalence classes**. For an element \( a otin A \), its equivalence class, denoted by \( [a] \), consists of all elements in A that are related to \( a \). \[ [a] = \{x otin A ext{ | } a R x\} \] ### Vocabulary * **Relation:** A set of ordered pairs showing a connection between elements from sets. * **Cartesian Product:** For sets A and B, denoted \( A imes B \), it is the set of all possible ordered pairs \( (a, b) \) where \( a otin A \) and \( b otin B \). * **Reflexive:** A relation where every element is related to itself. * **Symmetric:** A relation where if \( a \) is related to \( b \), then \( b \) is also related to \( a \). * **Transitive:** A relation where if \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is related to \( c \). * **Equivalence Relation:** A relation that is reflexive, symmetric, and transitive. * **Equivalence Class:** A subset of elements in a set that are all related to each other under an equivalence relation. ### Examples **Example 1: Identifying Properties** Let set \( A = \{1, 2, 3\} \) and consider the following relations on A: * \( R_1 = \{(1,1), (2,2), (3,3), (1,2)\} \) * **Reflexive?** Yes, because \( (1,1), (2,2), (3,3) \) are all in \( R_1 \). * **Symmetric?** No, because \( (1,2) otin R_1 \) but \( (2,1) otin R_1 \). * **Transitive?** Yes. The only case to check is \( (1,1) otin R_1 \) and \( (1,2) otin R_1 \), which implies \( (1,2) otin R_1 \) (which is true). No violations. * \( R_2 = \{(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)\} \) * **Reflexive?** Yes, all \( (a,a) \) pairs are present. * **Symmetric?** Yes. \( (1,2) \) has \( (2,1) \); \( (2,3) \) has \( (3,2) \). * **Transitive?** No. Consider \( (1,2) otin R_2 \) and \( (2,3) otin R_2 \). For transitivity, \( (1,3) \) must be in \( R_2 \). But \( (1,3) otin R_2 \). So, \( R_2 \) is not transitive. **Example 2: Real-World Relation** Consider the set of all students in your school. Let R be the relation "has the same height as". * **Reflexive?** Yes, every student has the same height as themselves. * **Symmetric?** Yes, if Student A has the same height as Student B, then Student B has the same height as Student A. * **Transitive?** Yes, if Student A has the same height as Student B, and Student B has the same height as Student C, then Student A has the same height as Student C. * **Conclusion:** Since the relation is reflexive, symmetric, and transitive, it is an **equivalence relation**. The equivalence classes would be groups of students who all share the exact same height. ### Exercises 1. Let \( A = \{a, b, c\} \) and \( R = \{(a,a), (b,b), (c,c), (a,b)\} \). Determine whether R is reflexive, symmetric, or transitive. 2. Consider the relation "is a factor of" on the set \( P = \{2, 3, 4, 6\} \). List the ordered pairs in this relation. Is it reflexive? Is it symmetric? Is it transitive? 3. For the set of integers \( Z \), determine which properties (reflexive, symmetric, transitive) the relation "is greater than or equal to" (\( ot \)) possesses. ### Assignment 1. Let \( S = \{x, y, z, w\} \). Define a relation \( R_A \) on S that is reflexive and symmetric but not transitive. List the ordered pairs for \( R_A \). 2. Given the relation \( R_B = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\} \) on the set \( \{1, 2, 3\} \). * Is \( R_B \) reflexive? * Is \( R_B \) symmetric? * Is \( R_B \) transitive? * Is \( R_B \) an equivalence relation? 3. Consider the relation "has the same number of letters as" on the set of words \( W = \{\text{cat, dog, table, chair, book}\}\) . * Is this relation reflexive? * Is this relation symmetric? * Is this relation transitive? * Is this an equivalence relation? If yes, identify at least two equivalence classes. ### Summary In this lesson, we explored the fundamental properties of relations on a set. We learned that a relation can be: * **Reflexive:** If every element relates to itself. * **Symmetric:** If \( a \) relates to \( b \), then \( b \) relates to \( a \). * **Transitive:** If \( a \) relates to \( b \) and \( b \) relates to \( c \), then \( a \) relates to \( c \). When a relation satisfies all three of these properties (reflexive, symmetric, and transitive), it is called an **equivalence relation**. Equivalence relations are very important in mathematics as they partition a set into distinct equivalence classes, grouping together elements that share a certain characteristic. Understanding these properties is crucial for further studies in various branches of mathematics. ## Equivalent relations. ### Introduction Welcome to our lesson on equivalent relations! In mathematics, a **relation** describes how elements in a set are connected or linked. We often encounter relations in our daily lives, like "is taller than," "is a friend of," or "is related to." In this lesson, we will explore a special type of relation called an **equivalent relation**. Understanding equivalent relations helps us group elements that share certain properties, which is a powerful concept in many areas of mathematics. ### Key Concepts #### What is a Relation? A relation \( R \) on a set \( A \) is simply a collection of ordered pairs \( (a, b) \) where \( a \) and \( b \) are elements of \( A \). If \( (a, b) \in R \), we say that \( a \) is related to \( b \), sometimes written as \( aRb \). **Example:** Let set \( A = \{1, 2, 3\} \). A relation \( R \) could be \( \{(1, 1), (1, 2), (2, 3)\} \). #### Properties of Relations For a relation to be an equivalence relation, it must satisfy three important properties: ##### 1. Reflexive Relation A relation \( R \) on a set \( A \) is **reflexive** if every element in \( A \) is related to itself. That is, for every \( a \in A \), \( (a, a) \in R \). **Example:** * The relation "is equal to" (\( = \)) on a set of numbers is reflexive because any number is equal to itself (e.g., \( 5 = 5 \)). * The relation "is less than" (\( < \)) is not reflexive because a number is not less than itself (e.g., \( 5 \not< 5 \)). ##### 2. Symmetric Relation A relation \( R \) on a set \( A \) is **symmetric** if whenever \( a \) is related to \( b \), then \( b \) is also related to \( a \). That is, if \( (a, b) \in R \), then \( (b, a) \in R \). **Example:** * The relation "is a sibling of" is symmetric: if Ali is a sibling of Ben, then Ben is a sibling of Ali. * The relation "is a parent of" is not symmetric: if John is the parent of Mary, Mary is not the parent of John. ##### 3. Transitive Relation A relation \( R \) on a set \( A \) is **transitive** if whenever \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is also related to \( c \). That is, if \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \). **Example:** * The relation "is less than" (\( < \)) is transitive: if \( a < b \) and \( b < c \), then \( a < c \). * The relation "is a friend of" is not necessarily transitive: if Ali is a friend of Ben, and Ben is a friend of Cindy, it doesn't automatically mean Ali is a friend of Cindy. #### Equivalent Relation A relation \( R \) on a set \( A \) is an **equivalence relation** if it is: 1. **Reflexive** 2. **Symmetric** 3. **Transitive** Equivalence relations are crucial because they partition a set into disjoint (non-overlapping) subsets called equivalence classes. #### Equivalence Classes If \( R \) is an equivalence relation on a set \( A \) and \( a \in A \), the **equivalence class** of \( a \), denoted by \( [a] \) (or sometimes \( \bar{a} \)), is the set of all elements in \( A \) that are related to \( a \). \[ [a] = \{x \in A \mid (x, a) \in R\} \] **Properties of Equivalence Classes:** * Every element \( a \in A \) belongs to exactly one equivalence class. * Two equivalence classes \( [a] \) and \( [b] \) are either identical (if \( a \) and \( b \) are related) or disjoint (if \( a \) and \( b \) are not related). * The set of all equivalence classes forms a **partition** of the set \( A \). ### Vocabulary * **Relation:** A connection between elements of a set, expressed as a set of ordered pairs. * **Reflexive Relation:** A relation where every element is related to itself (\( aRa \)). * **Symmetric Relation:** A relation where if \( a \) is related to \( b \), then \( b \) is related to \( a \) (if \( aRb \), then \( bRa \)). * **Transitive Relation:** A relation where if \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is related to \( c \) (if \( aRb \) and \( bRc \), then \( aRc \)). * **Equivalent Relation:** A relation that is reflexive, symmetric, and transitive. * **Equivalence Class:** A subset of elements in a set that are all related to each other by an equivalence relation. * **Partition:** A division of a set into non-empty, non-overlapping subsets whose union is the original set. ### Examples #### Example 1: The "is equal to" relation Let \( A \) be the set of all integers, \( A = \mathbb{Z} \). Consider the relation \( R \) defined by \( aRb \) if \( a = b \). 1. **Reflexive:** For any integer \( a \in \mathbb{Z} \), \( a = a \). So, \( (a, a) \in R \). The relation is reflexive. 2. **Symmetric:** If \( a = b \), then it is also true that \( b = a \). So, if \( (a, b) \in R \), then \( (b, a) \in R \). The relation is symmetric. 3. **Transitive:** If \( a = b \) and \( b = c \), then it must be that \( a = c \). So, if \( (a, b) \in R \) and \( (b, c) \in R \), then \( (a, c) \in R \). The relation is transitive. Since \( R \) is reflexive, symmetric, and transitive, it is an equivalence relation. The equivalence class of any integer \( a \) is just \( [a] = \{a\} \), meaning each number forms its own class. #### Example 2: The "has the same number of letters as" relation Let \( A \) be the set of words \( \{\text{cat, dog, bird, elephant, mouse}\} \). Define a relation \( R \) on \( A \) such that \( w_1 R w_2 \) if \( w_1 \) has the same number of letters as \( w_2 \). Let's check the properties: 1. **Reflexive:** Does every word have the same number of letters as itself? Yes, "cat" has 3 letters, and "cat" has 3 letters. So, \( (w, w) \in R \) for all \( w \in A \). It is reflexive. 2. **Symmetric:** If "cat" has the same number of letters as "dog" (3 letters each), then "dog" also has the same number of letters as "cat". So, if \( (w_1, w_2) \in R \), then \( (w_2, w_1) \in R \). It is symmetric. 3. **Transitive:** If "cat" has the same number of letters as "dog" (3 letters), and "dog" has the same number of letters as "hen" (3 letters), then "cat" also has the same number of letters as "hen". So, if \( (w_1, w_2) \in R \) and \( (w_2, w_3) \in R \), then \( (w_1, w_3) \in R \). It is transitive. Since it is reflexive, symmetric, and transitive, this is an equivalence relation. **Equivalence Classes:** * Words with 3 letters: \( [\text{cat}] = [\text{dog}] = \{\text{cat, dog}\} \) * Words with 4 letters: \( [\text{bird}] = \{\text{bird}\} \) * Words with 5 letters: \( [\text{mouse}] = \{\text{mouse}\} \) * Words with 8 letters: \( [\text{elephant}] = \{\text{elephant}\} \) #### Example 3: The "is parallel to" relation Let \( L \) be the set of all lines in a plane. Define a relation \( R \) on \( L \) such that \( l_1 R l_2 \) if line \( l_1 \) is parallel to line \( l_2 \) (denoted \( l_1 \parallel l_2 \)). Assume a line is parallel to itself. Let's check the properties: 1. **Reflexive:** Is every line parallel to itself? Yes, by definition in geometry, \( l \parallel l \). So, \( (l, l) \in R \). It is reflexive. 2. **Symmetric:** If line \( l_1 \) is parallel to line \( l_2 \) (i.e., \( l_1 \parallel l_2 \)), is line \( l_2 \) parallel to line \( l_1 \) (i.e., \( l_2 \parallel l_1 \))? Yes. So, if \( (l_1, l_2) \in R \), then \( (l_2, l_1) \in R \). It is symmetric. 3. **Transitive:** If line \( l_1 \) is parallel to line \( l_2 \) (\( l_1 \parallel l_2 \)), and line \( l_2 \) is parallel to line \( l_3 \) (\( l_2 \parallel l_3 \)), then is line \( l_1 \) parallel to line \( l_3 \) (\( l_1 \parallel l_3 \))? Yes, this is a property of parallel lines. So, if \( (l_1, l_2) \in R \) and \( (l_2, l_3) \in R \), then \( (l_1, l_3) \in R \). It is transitive. Since the relation is reflexive, symmetric, and transitive, "is parallel to" is an equivalence relation. The equivalence classes would consist of all lines that are parallel to each other. ### Exercises 1. Let \( A = \{1, 2, 3, 4\} \) and \( R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 4), (4, 3)\} \). * Is \( R \) reflexive? Explain. * Is \( R \) symmetric? Explain. * Is \( R \) transitive? Explain. * Is \( R \) an equivalence relation? If so, find its equivalence classes. 2. Consider the relation \( R \) on the set of all students in your class, where \( aRb \) if \( a \) and \( b \) are in the same mathematics group. * Is this relation reflexive? Why? * Is this relation symmetric? Why? * Is this relation transitive? Why? * Is it an equivalence relation? What would the equivalence classes represent? 3. On the set of all triangles, define a relation \( \sim \) such that \( T_1 \sim T_2 \) if \( T_1 \) is congruent to \( T_2 \). * Check if this relation is reflexive, symmetric, and transitive. State your reasons. * Is it an equivalence relation? If yes, what do the equivalence classes represent? 4. Let \( A = \{a, b, c\} \). Give an example of a relation on \( A \) that is: * Reflexive and symmetric, but not transitive. * Symmetric and transitive, but not reflexive. * Reflexive and transitive, but not symmetric. ### Assignment 1. On the set of integers \( \mathbb{Z} \), define a relation \( R \) such that \( aRb \) if \( a - b \) is an even number. * Prove that \( R \) is an equivalence relation. * Find the equivalence classes \( [0] \) and \( [1] \). Describe these classes in words. 2. Let \( A = \{P_1, P_2, P_3, P_4\} \) be a set of four different people. Define a relation \( R = \{(P_1, P_1), (P_2, P_2), (P_3, P_3), (P_4, P_4), (P_1, P_2), (P_2, P_1), (P_3, P_4), (P_4, P_3)\} \). Determine if \( R \) is an equivalence relation. If it is, list all the equivalence classes. 3. Consider the relation "is a factor of" on the set of positive integers \( \mathbb{Z}^+ \). Is this relation reflexive, symmetric, or transitive? Is it an equivalence relation? Justify your answers. ### Summary Today, we learned about **equivalent relations**, which are a fundamental concept in mathematics. An equivalence relation is a special type of relation that must satisfy three key properties: * **Reflexive:** Every element is related to itself (e.g., \( a = a \)). * **Symmetric:** If \( a \) is related to \( b \), then \( b \) is related to \( a \) (e.g., if \( a \parallel b \), then \( b \parallel a \)). * **Transitive:** If \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is related to \( c \) (e.g., if \( a < b \) and \( b < c \), then \( a < c \)). When a relation is all three, it allows us to group elements into **equivalence classes**, where all elements within a class are related to each other. These classes form a **partition** of the original set, dividing it into distinct, non-overlapping groups. Understanding these properties helps us analyze and categorize mathematical objects more effectively. ## Domain, Codomain, Image and Range. # Domain, Codomain, Image and Range.\n\n## 1. Introduction\nWelcome to this lesson on Relations and Functions! In mathematics, functions are like machines that take an input and give a specific output. To truly understand how these machines work, we need to define exactly what inputs they can take and what outputs they can produce. This lesson will introduce you to four important terms that help us do just that: Domain, Codomain, Image, and Range.\n\n## 2. Key Concepts\n\n### What is a Relation?\nA **relation** is simply a set of ordered pairs. For example, \\( \{(1, A), (2, B), (3, A)\} \\) is a relation. It shows how elements from one set are related to elements from another set.\n\n### What is a Function?\nA **function** is a special type of relation where each input has exactly one output. This means that for any value you put into the function, you will always get one and only one specific output. If an input has more than one output, it is not a function.\n\n### 2.1. Domain\n* **Definition:** The **Domain** is the set of all possible input values (often represented by \\( x \\) values) for which a function or relation is defined. Think of it as all the ingredients you are allowed to put into your 'function machine'.\n* **Notation:** The domain is usually denoted by \\( D_f \\) or just \\( D \\).\n* **How to find it:**\n * **From a set of ordered pairs:** It's the set of all first elements in the ordered pairs.\n * **From an equation:** Consider values of \\( x \\) that won't make the function undefined (e.g., division by zero, square root of a negative number).\n * **From a graph:** Look at the \\( x \\)-values covered by the graph from left to right.\n\n### 2.2. Codomain\n* **Definition:** The **Codomain** is the set of all possible output values that the function *could* potentially produce. It is the target set that contains all the possible images. It's usually specified when a function is defined as \\( f: A \to B \\), where \\( A \\) is the domain and \\( B \\) is the codomain.\n* **Distinction from Range:** The codomain is the *potential* set of outputs, while the range is the set of *actual* outputs.\n\n### 2.3. Image\n* **Definition:** The **Image** of a specific input value (say, \\( x \\)) under a function \\( f \\) is the actual output value, \\( f(x) \\). If \\( f(x) = y \\), then \\( y \\) is the image of \\( x \\).\n* **Example:** If \\( f(x) = x+1 \\), then the image of \\( 2 \\) is \\( f(2) = 2+1 = 3 \\).\n\n### 2.4. Range\n* **Definition:** The **Range** is the set of all actual output values (often represented by \\( y \\) values) that a function or relation produces for the given domain. It is a subset of the codomain. Think of it as all the actual products your 'function machine' can make.\n* **Notation:** The range is usually denoted by \\( R_f \\) or just \\( R \\).\n* **How to find it:**\n * **From a set of ordered pairs:** It's the set of all second elements in the ordered pairs.\n * **From an equation:** Determine all possible \\( y \\) values resulting from the domain.\n * **From a graph:** Look at the \\( y \\)-values covered by the graph from bottom to top.\n\n## 3. Vocabulary\n* **Relation:** A set of ordered pairs showing a connection between elements of two sets.\n* **Function:** A special relation where each input has exactly one output.\n* **Ordered Pair:** A pair of numbers \\( (x, y) \\) where \\( x \\) is the input and \\( y \\) is the output.\n* **Input:** The value given to a function (x-value).\n* **Output:** The value produced by a function (y-value).\n* **Domain:** The set of all possible input values.\n* **Codomain:** The set of all possible output values as specified for the function.\n* **Image:** The specific output value for a given input value.\n* **Range:** The set of all *actual* output values that a function produces.\n\n## 4. Examples\n\n### Example 1: Using a Set of Ordered Pairs\nConsider the relation \\( R = \{(1, 2), (2, 4), (3, 6), (4, 4)\} \\).\n* **Domain:** The set of all first elements: \\( \{1, 2, 3, 4\} \\).\n* **Codomain:** If not explicitly stated, we can assume it's the set of all numbers encountered or a larger set like integers or real numbers, depending on context. Let's assume the codomain is \\( \{1, 2, 3, 4, 5, 6, 7\} \\).\n* **Image:**\n * The image of 1 is 2.\n * The image of 2 is 4.\n * The image of 3 is 6.\n * The image of 4 is 4.\n* **Range:** The set of all second elements: \\( \{2, 4, 6\} \\). Notice that 4 appears twice as an image, but it's listed only once in the range set. Also, the range \\( \{2, 4, 6\} \\) is a subset of the codomain \\( \{1, 2, 3, 4, 5, 6, 7\} \\).\n\n### Example 2: Using a Function Equation\nLet \\( f(x) = 2x + 1 \\) with the domain \\( D = \{-1, 0, 1, 2\} \\). The codomain is specified as the set of Integers (\\( \mathbb{Z} \\)).\n* **Domain:** \\( \{-1, 0, 1, 2\} \\) (given).\n* **Codomain:** \\( \mathbb{Z} \\) (given).\n* **Images:**\n * For \\( x = -1 \\), \\( f(-1) = 2(-1) + 1 = -2 + 1 = -1 \\). The image of -1 is -1.\n * For \\( x = 0 \\), \\( f(0) = 2(0) + 1 = 0 + 1 = 1 \\). The image of 0 is 1.\n * For \\( x = 1 \\), \\( f(1) = 2(1) + 1 = 2 + 1 = 3 \\). The image of 1 is 3.\n * For \\( x = 2 \\), \\( f(2) = 2(2) + 1 = 4 + 1 = 5 \\). The image of 2 is 5.\n* **Range:** The set of all actual output values: \\( \{-1, 1, 3, 5\} \\). This set is indeed a subset of the integers.\n\n### Example 3: Finding Domain and Range from an Equation (Real Numbers)\nConsider the function \\( g(x) = \sqrt{x-3} \\).\n* **Domain:** For \\( g(x) \\) to be defined in real numbers, the expression inside the square root must be non-negative. \\( x-3 \ge 0 \implies x \ge 3 \\). So, the Domain is \\( \{x \in \mathbb{R} \mid x \ge 3\} \\).\n* **Range:** Since \\( \sqrt{x-3} \\) will always produce non-negative values, the Range is \\( \{y \in \mathbb{R} \mid y \ge 0\} \\).\n\n## 5. Exercises\nFor each of the following, identify the Domain, Codomain (if explicitly stated, otherwise assume \\( \mathbb{R} \\)), the Image of specific values, and the Range.\n\n1. Relation \\( P = \{(a, X), (b, Y), (c, X), (d, Z)\} \\). Assume codomain is \\( \{X, Y, Z, W\} \\).\n * What is the domain?\n * What is the image of \\( b \\)?\n * What is the range?\n\n2. Function \\( h(x) = x^2 - 3 \\) with domain \\( D = \{-3, -1, 0, 1, 3\} \\). Codomain is \\( \mathbb{Z} \\).\n * Find the image of \\( -1 \\) and \\( 3 \\).\n * Determine the range of \\( h(x) \\).\n\n3. For the function \\( k(x) = \frac{1}{x+2} \\), where \\( x \\) is a real number.\n * What is the domain of \\( k(x) \\)?\n * What is the range of \\( k(x) \\)?\n\n## 6. Assignment\n1. Given the function \\( f(x) = \frac{x-1}{x^2 - 4} \\).\n * Determine the domain of \\( f(x) \\).\n * Find the image of \\( x=1 \\) and \\( x=3 \\).\n\n2. A company's profit \\( P \\) (in thousands of dollars) is a function of the number of items \\( n \\) sold, given by \\( P(n) = 5n - 100 \\).\n * If the company sells between 30 and 200 items (inclusive), what is the domain for this function?\n * What is the range of profit for this domain?\n\n3. Sketch the graph of \\( y = 2x - 3 \\) for \\( -2 \le x \le 3 \\). From your graph, identify the domain and range.\n\n## 7. Summary\n* The **Domain** is the set of all possible input values.\n* The **Codomain** is the set of all *potential* output values, a target set defined for the function.\n* The **Image** is the specific output value for a particular input.\n* The **Range** is the set of all *actual* output values produced by the function, and it is always a subset of the Codomain.\nUnderstanding these terms is fundamental to working with functions and interpreting their behavior correctly. Always remember to consider the specified domain when determining the range of a function. ## Types of mappings ### Introduction Welcome to this lesson on "Types of Mappings"! In mathematics, we often explore relationships between different sets of numbers or objects. A **mapping** is a fundamental concept that describes how elements from one set are related to elements in another set. Understanding different types of mappings is crucial for grasping more advanced mathematical concepts like functions and transformations. In this lesson, we will define what a mapping is, identify its key components, and then delve into the four main types of mappings: one-to-one, many-to-one, one-to-many, and many-to-many. We'll also revisit how functions fit into this classification. ### Key Concepts #### What is a Mapping? A mapping (also known as a relation) is a rule that associates each element from a set, called the **domain**, with one or more elements in another set, called the **codomain**. Think of it like a machine that takes an input and produces an output. Mathematically, if \( A \) and \( B \) are two non-empty sets, a mapping \( f \) from \( A \) to \( B \) associates each element \( x \in A \) with at least one element \( y \in B \). #### Components of a Mapping * **Domain:** The set of all possible input values (or objects) for the mapping. Each element in the domain must be mapped to at least one element in the codomain. * **Codomain:** The set of all possible output values (or images) for the mapping. It's the set where the mapped elements reside. * **Range:** The subset of the codomain that contains all the actual output values (images) produced by the mapping. In other words, the range consists of all elements in the codomain that are mapped to by at least one element from the domain. * **Object:** An element from the domain. * **Image:** An element from the codomain that is mapped from an object in the domain. #### Types of Mappings There are four primary types of mappings, classified by how elements in the domain relate to elements in the codomain. ##### 1. One-to-one Mapping (Injection) In a one-to-one mapping, each distinct element in the domain is mapped to a distinct element in the codomain. No two elements in the domain map to the same element in the codomain. * **Characteristic:** Every object has exactly one unique image. * **Example Diagram:** * `A = {1, 2, 3}` * `B = {a, b, c, d}` * Mapping: `{(1, a), (2, b), (3, c)}` ##### 2. Many-to-one Mapping In a many-to-one mapping, two or more different elements in the domain are mapped to the same single element in the codomain. * **Characteristic:** Multiple objects can share the same image, but each object still has only one image. * **Example Diagram:** * `A = {-2, -1, 1, 2}` * `B = {1, 4}` * Mapping: `{( -2, 4 ), ( -1, 1 ), ( 1, 1 ), ( 2, 4 )}` (e.g., \( f(x) = x^2 \)) ##### 3. One-to-many Mapping In a one-to-many mapping, a single element in the domain is mapped to two or more different elements in the codomain. * **Characteristic:** One object has multiple images. * **Example Diagram:** * `A = {4}` * `B = {-2, 2}` * Mapping: `{(4, -2), (4, 2)}` (e.g., \( x \rightarrow \pm \sqrt{x} \)) ##### 4. Many-to-many Mapping In a many-to-many mapping, at least one element in the domain is mapped to more than one element in the codomain, and at least one element in the codomain is mapped from more than one element in the domain. * **Characteristic:** Multiple objects can have multiple images, and multiple images can be associated with multiple objects. * **Example Diagram:** * `A = {1, 2}` * `B = {a, b, c}` * Mapping: `{(1, a), (1, b), (2, b), (2, c)}` #### Functions A **function** is a special type of mapping where every element in the domain is mapped to *exactly one* element in the codomain. This means that **one-to-one mappings** and **many-to-one mappings** are functions. However, **one-to-many** and **many-to-many** mappings are *not* functions because at least one element in their domain has more than one image. ### Vocabulary * **Mapping (Relation):** A rule that associates elements from a domain to elements in a codomain. * **Domain:** The set of all input values or objects. * **Codomain:** The set of all possible output values or images. * **Range:** The subset of the codomain containing all actual output values produced by the mapping. * **Object:** An element in the domain. * **Image:** An element in the codomain that is mapped from an object. * **One-to-one Mapping (Injection):** Each distinct object maps to a distinct image. * **Many-to-one Mapping:** Multiple objects map to the same image. * **One-to-many Mapping:** One object maps to multiple images. * **Many-to-many Mapping:** Multiple objects map to multiple images. * **Function:** A mapping where each object in the domain has exactly one image in the codomain (i.e., one-to-one or many-to-one mappings). ### Examples #### Example 1: Real-world Scenario - Students and ID Numbers Consider a school where each student is assigned a unique student ID number. * **Domain:** Set of all students. * **Codomain:** Set of all possible student ID numbers. * **Mapping:** Assigning an ID to a student. * **Type:** This is a **one-to-one mapping** because each student has only one ID, and each ID belongs to only one student. #### Example 2: Real-world Scenario - Students and Subjects Consider students in a class and the subjects they take. * **Domain:** Set of all students. * **Codomain:** Set of all subjects offered. * **Mapping:** Student \( \rightarrow \) Subject(s) they take. * If student A takes Math, Science, and English, and student B takes Math and History, and student C takes Science. * **Type:** This is a **many-to-many mapping**. Student A takes multiple subjects (one-to-many aspect from the domain perspective). Math is taken by student A and B (many-to-one aspect from the codomain perspective). Since both occur, it's many-to-many. #### Example 3: Algebraic Mapping Let \( f(x) = x^2 \) for \( x \in \{-3, -2, -1, 0, 1, 2, 3\} \). * **Domain:** \( D = \{-3, -2, -1, 0, 1, 2, 3\} \) * **Codomain:** Let's say \( C = \{0, 1, 4, 9, 16\} \) * **Mappings:** * \( -3 \rightarrow 9 \) * \( -2 \rightarrow 4 \) * \( -1 \rightarrow 1 \) * \( 0 \rightarrow 0 \) * \( 1 \rightarrow 1 \) * \( 2 \rightarrow 4 \) * \( 3 \rightarrow 9 \) * **Type:** This is a **many-to-one mapping** because, for instance, both \( -1 \) and \( 1 \) map to \( 1 \), and both \( -2 \) and \( 2 \) map to \( 4 \). It is a function because each input has exactly one output. ### Exercises Determine the type of mapping for each of the following relations and state whether it is a function: 1. Given sets \( A = \{p, q, r\} \) and \( B = \{x, y, z\} \). The mapping is \( \{(p, x), (q, y), (r, z)\} \). 2. Given sets \( C = \{1, 2, 3\} \) and \( D = \{a, b\} \). The mapping is \( \{(1, a), (2, b), (3, a)\} \). 3. Given sets \( E = \{4\} \) and \( F = \{s, t, u\} \). The mapping is \( \{(4, s), (4, t)\} \). 4. Given sets \( G = \{apple, banana\} \) and \( H = \{red, yellow, green\} \). The mapping is \( \{(apple, red), (apple, green), (banana, yellow), (banana, green)\} \). 5. The relation \( y = x^3 \) for \( x \in \mathbb{R} \). 6. The relation \( x = y^2 \) for \( y \in \mathbb{R} \). ### Assignment 1. For each of the following, define the domain and codomain, then identify the type of mapping and state if it is a function: a. People and their fingerprints. b. Books in a library and their authors (assume a book can have multiple authors, and an author can write multiple books). c. Numbers and their absolute values (e.g., \( |-5| = 5 \) and \( |5| = 5 \)). d. The relation \( f(x) = \sqrt{x} \) for \( x \ge 0 \). 2. Draw a diagram representing a many-to-many mapping between two sets, \( P \) and \( Q \), each having at least 3 elements. 3. Explain why one-to-many and many-to-many mappings are not considered functions. ### Summary In this lesson, we explored the concept of mappings, which describe relationships between elements of two sets: the domain and the codomain. We learned about the range, which is the set of all actual output values. We identified four main types of mappings: * **One-to-one:** Each unique input maps to a unique output. * **Many-to-one:** Multiple inputs map to the same output. * **One-to-many:** A single input maps to multiple outputs. * **Many-to-many:** Multiple inputs map to multiple outputs. Crucially, we distinguished between general mappings and **functions**. A function is a specific type of mapping where every element in the domain maps to *exactly one* element in the codomain. Therefore, only one-to-one and many-to-one mappings are considered functions. Understanding these distinctions is fundamental to your journey in higher mathematics. ## Flow diagrams # Flow diagrams\n\n## Introduction\nIn mathematics, understanding how numbers are transformed by operations is crucial. Flow diagrams provide a visual and step-by-step way to represent these transformations, especially useful when dealing with relations and functions. They help us see clearly how an input value goes through a sequence of operations to become an output value.\n\n## Key Concepts\n\n### What is a Flow Diagram?\nA flow diagram is a visual representation that illustrates a sequence of operations applied to an input value to produce an output value. It uses arrows to show the direction of the process and boxes to indicate operations.\n\n### Components of a Flow Diagram\nEvery flow diagram consists of:\n* **Input:** The starting value that enters the diagram.\n* **Process/Rule:** The mathematical operation(s) or instruction(s) applied to the input. These are usually written inside boxes. Examples include "add 5", "multiply by 2", "subtract 3", "divide by 4".\n* **Output:** The resulting value after the input has passed through all the processes.\n* **Arrows:** These connect the input, processes, and output, showing the direction of the flow.\n\n### Constructing a Flow Diagram\nTo construct a flow diagram for a given rule or expression, follow these steps:\n1. Start with the input variable (e.g., \\( x \\)).\n2. Identify the first operation applied to the input and place it in a box, connected by an arrow.\n3. Identify the next operation applied to the result of the previous step and place it in another box, connected by an arrow.\n4. Continue this until all operations in the rule have been applied.\n5. The final result is the output variable (e.g., \\( y \\)).\n\n**Example:** For the rule \\( y = 2x + 3 \\)\nInput: \\( x \\)\nFirst process: Multiply by 2 (\\( 2x \\))\nSecond process: Add 3 (\\( 2x + 3 \\))\nOutput: \\( y \\)\n\n### Representing Relations and Functions\nFlow diagrams are excellent tools for representing relations and functions because they clearly show the relationship between the input and the output. Each process box represents a specific mathematical operation.\n\n### Inverse Operations in Flow Diagrams\nAn inverse operation "undoes" the original operation. For example:\n* The inverse of addition is subtraction.\n* The inverse of multiplication is division.\n* The inverse of subtraction is addition.\n* The inverse of division is multiplication.\n\nTo find the input value when the output value is known, we can reverse the flow diagram. This involves:\n1. Starting from the output value.\n2. Working backwards through the diagram.\n3. Applying the inverse operation for each process box, in reverse order.\n\n**Example:** If a flow diagram has "multiply by 2" then "add 3", the inverse flow diagram would have "subtract 3" then "divide by 2".\n\n## Vocabulary\n* **Flow Diagram:** A visual representation of a sequence of operations transforming an input to an output.\n* **Input:** The initial value fed into a process or system.\n* **Output:** The resulting value produced by a process or system.\n* **Process:** A specific operation or set of operations performed on a value.\n* **Rule:** The mathematical instruction that defines how an input is transformed into an output.\n* **Relation:** A set of ordered pairs showing a relationship between elements of two sets.\n* **Function:** A special type of relation where each input has exactly one output.\n* **Inverse Operation:** An operation that reverses the effect of another operation.\n\n## Examples\n\n### Example 1: Constructing a Flow Diagram and Finding Output\n**Rule:** \\( y = 3x - 4 \\)\n**a) Construct the flow diagram.**\nInput: \\( x \\) \\(\rightarrow\\) [Multiply by 3] \\(\rightarrow\\) [Subtract 4] \\(\rightarrow\\) Output: \\( y \\)\n\n**b) Find the output when the input is 5.**\nStarting with input 5:\n\\( 5 \\xrightarrow{\\text{Multiply by 3}} 15 \\xrightarrow{\\text{Subtract 4}} 11 \\)\nSo, when \\( x=5 \\), \\( y=11 \\).\n\n### Example 2: Working Backwards with Inverse Operations\n**Flow Diagram:** \\( x \\rightarrow \\text{[Add 7]} \\rightarrow \\text{[Divide by 2]} \\rightarrow y \\)\n**Find the input when the output is 10.**\nTo find the input, we reverse the diagram and use inverse operations:\nOutput: \\( 10 \\) \\(\leftarrow\\) [Multiply by 2] \\(\leftarrow\\) [Subtract 7] \\(\leftarrow\\) Input: \\( x \\)\n\n1. Start with the output: \\( 10 \\)\n2. Apply the inverse of "Divide by 2", which is "Multiply by 2": \\( 10 \\times 2 = 20 \\)\n3. Apply the inverse of "Add 7", which is "Subtract 7": \\( 20 - 7 = 13 \\)\nSo, when \\( y=10 \\), the input \\( x=13 \\).\n\n## Exercises\n\n1. Construct a flow diagram for each of the following rules:\n a) \\( y = x + 8 \\)\n b) \\( y = 5x \\)\n c) \\( y = \\frac{x}{3} - 2 \\)\n d) \\( y = 2(x - 1) \\)\n\n2. For the flow diagram \\( x \\rightarrow \\text{[Multiply by 4]} \\rightarrow \\text{[Add 1]} \\rightarrow y \\):\n a) Find the output when the input is 3.\n b) Find the output when the input is -2.\n c) Find the input when the output is 13.\n\n3. Given the flow diagram: Input \\( k \\) \\(\rightarrow\\) [Subtract 5] \\(\rightarrow\\) [Multiply by 6] \\(\rightarrow\\) Output \\( m \\).\n a) Write down the rule for this flow diagram.\n b) Find \\( m \\) when \\( k = 7 \\).\n c) Find \\( k \\) when \\( m = 18 \\).\n\n## Assignment\n\n1. Draw the flow diagram for the rule \\( p = \\frac{q+10}{2} \\).\n2. For the flow diagram \\( x \\rightarrow \\text{[Add 9]} \\rightarrow \\text{[Divide by 3]} \\rightarrow \\text{[Subtract 1]} \\rightarrow y \\):\n a) Calculate the output when the input is 12.\n b) Determine the input when the output is 4.\n3. A tailor charges \\( R_s \\) 200 for stitching a shirt and an additional \\( R_s \\) 50 per meter of fabric used.\n a) If \\( F \\) is the number of meters of fabric used, write a rule for the total cost \\( C \\) in terms of \\( F \\).\n b) Draw a flow diagram to represent this rule.\n c) Use your flow diagram to find the total cost if 3 meters of fabric are used.\n d) If a customer paid \\( R_s \\) 350, how many meters of fabric were used?\n\n## Summary\nFlow diagrams are powerful visual tools for understanding how mathematical operations transform numbers. They clearly show the path from an **input** to an **output** through a sequence of **processes** or **rules**. By understanding how to construct and interpret these diagrams, you can easily represent functions and relations. Furthermore, the concept of **inverse operations** allows us to work backwards through a flow diagram to find an unknown input from a known output, which is a fundamental skill in algebra. ## Functions ### Introduction Welcome to our lesson on functions! In mathematics, we often deal with relationships between different quantities. A **function** is a special type of relationship where each input has exactly one output. This unique correspondence makes functions a fundamental concept in mathematics and various fields of science and engineering. Think about a vending machine: when you press a button (input), you get one specific snack or drink (output). You wouldn't expect pressing one button to give you two different items, or sometimes one item and sometimes another. That's the essence of a function – predictability and a single, clear outcome for every action. ### Key Concepts #### 1. What is a Function? A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. * **Relation:** A set of ordered pairs \( (x, y) \). It simply shows how elements from one set are connected to elements from another set. * **Function:** A specific type of relation where for every \( x \) value (input), there is only one \( y \) value (output). #### 2. Components of a Function * **Domain:** The set of all possible input values (\( x \)-values) for which the function is defined. It's what you 'feed' into the function. * **Codomain:** The set of all possible output values that *could* result from a function. It's the larger set from which outputs are drawn. * **Range:** The set of all *actual* output values (\( y \)-values) that the function produces. The range is a subset of the codomain. * **Independent Variable:** The input variable, typically \( x \), whose value can be chosen freely. * **Dependent Variable:** The output variable, typically \( y \), whose value depends on the independent variable. #### 3. Representing Functions Functions can be represented in several ways: * **a) Arrow Diagrams:** An arrow diagram shows how each element in the domain maps to an element in the codomain. For it to be a function, each element in the domain must have exactly one arrow originating from it. * **One-to-one function:** Each element in the domain maps to a unique element in the codomain. No two different inputs share the same output. * **Many-to-one function:** Multiple elements in the domain map to the same element in the codomain. * **One-to-many relation (NOT a function):** One element in the domain maps to more than one element in the codomain. * **b) Ordered Pairs:** A set of ordered pairs \( (x, y) \) where no two ordered pairs have the same first element (\( x \)-value) but different second elements (\( y \)-values). * **c) Tables:** A table can list input and corresponding output values. Each input \( x \) must have only one output \( y \). * **d) Graphs:** When a function is plotted on a coordinate plane, it must pass the **Vertical Line Test**. If any vertical line intersects the graph at more than one point, then the graph does not represent a function. * **e) Algebraic Expressions (Function Notation):** Functions are often expressed using algebraic formulas. The notation \( f(x) \) (read as "f of x") is used to represent the output of the function \( f \) when the input is \( x \). For example, if a function doubles a number and adds one, we write it as \( f(x) = 2x + 1 \). #### 4. Evaluating Functions To evaluate a function \( f(x) \) at a specific value, substitute that value for \( x \) in the function's expression. For example, if \( f(x) = 2x + 1 \), then \( f(3) \) means replace \( x \) with 3: \[ f(3) = 2(3) + 1 = 6 + 1 = 7 \] ### Vocabulary * **Function:** A relation where each input has exactly one output. * **Relation:** A set of ordered pairs. * **Domain:** The set of all possible input values. * **Codomain:** The set of all potential output values. * **Range:** The set of all actual output values produced by a function. * **Independent Variable:** The input variable (usually \( x \)). * **Dependent Variable:** The output variable (usually \( y \) or \( f(x) \)). * **One-to-one function:** Each input maps to a unique output. * **Many-to-one function:** Multiple inputs map to the same output. * **Vertical Line Test:** A graphical method to determine if a relation is a function. ### Examples **Example 1: Identifying a Function from Ordered Pairs** Which of the following sets of ordered pairs represents a function? a) \( \{ (1, 2), (2, 3), (3, 4), (4, 5) \} \) b) \( \{ (1, 2), (1, 3), (2, 4), (3, 5) \} \) c) \( \{ (-1, 0), (0, 0), (1, 0) \} \) * **Solution:** a) This is a function because each \( x \)-value (1, 2, 3, 4) appears only once as a first element. b) This is NOT a function because the \( x \)-value 1 is paired with two different \( y \)-values (2 and 3). c) This is a function (specifically, a many-to-one function) because each \( x \)-value (-1, 0, 1) appears only once, even though multiple \( x \)-values map to the same \( y \)-value (0). **Example 2: Identifying a Function from an Arrow Diagram** Consider the following diagram: ```mermaid graph TD A[1] --> X[a] B[2] --> Y[b] C[3] --> Z[c] ``` Is this a function? What are the domain and range? * **Solution:** Yes, this is a one-to-one function because each element in the domain (\( \{1, 2, 3\} \)) maps to exactly one unique element in the codomain. The domain is \( \{1, 2, 3\} \) and the range is \( \{a, b, c\} \). **Example 3: Evaluating a Function** Given the function \( f(x) = 3x - 5 \), find: a) \( f(2) \) b) \( f(-1) \) c) \( f(0) \) * **Solution:** a) \( f(2) = 3(2) - 5 = 6 - 5 = 1 \) b) \( f(-1) = 3(-1) - 5 = -3 - 5 = -8 \) c) \( f(0) = 3(0) - 5 = 0 - 5 = -5 \) **Example 4: Real-World Application** The cost \( C \) (in dollars) of producing \( n \) t-shirts is given by the function \( C(n) = 5n + 20 \). a) What is the cost of producing 10 t-shirts? b) What is the fixed cost (cost when no t-shirts are produced)? * **Solution:** a) To find the cost of 10 t-shirts, we evaluate \( C(10) \): \[ C(10) = 5(10) + 20 = 50 + 20 = 70 \] The cost is $70. b) The fixed cost is when \( n = 0 \): \[ C(0) = 5(0) + 20 = 0 + 20 = 20 \] The fixed cost is $20. ### Exercises 1. Determine whether each of the following relations is a function. State your reason. a) \( \{ (2, 4), (3, 6), (4, 8), (5, 10) \} \) b) \( \{ (0, 1), (0, 2), (1, 3), (2, 4) \} \) c) \( \{ (-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4) \} \) 2. Given the function \( g(x) = x^2 - 3 \), find: a) \( g(4) \) b) \( g(-2) \) c) \( g(0) \) 3. For the function \( h(x) = rac{1}{2}x + 7 \), find: a) \( h(10) \) b) The value of \( x \) for which \( h(x) = 15 \). 4. Sketch a graph of a relation that is a function and another that is not a function. Explain how the Vertical Line Test helps identify them. ### Assignment 1. Given the function \( f(x) = 5x - 8 \): a) Find \( f(3) \). b) Find \( f(-4) \). c) Find the value of \( x \) such that \( f(x) = 17 \). 2. A taxi charges a flat fee of $3 plus $2 for every kilometre traveled. Let \( K \) be the number of kilometres traveled and \( C(K) \) be the total cost. a) Write a function for \( C(K) \) in terms of \( K \). b) Calculate the cost of a 15-kilometre journey. c) If a journey cost $25, how many kilometres were traveled?