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Mathematics Chapter 1: Relations and Functions
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Mathematics Chapter 1: Relations and Functions

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Questions and Answers

What is the subset of A × A that represents a relation in a set A?

  • A × A
  • φ
  • A
  • All of the above (correct)

    • The relation R = {(a, b) : a – b = 10} in the set A = {1, 2, 3, 4} is the universal relation.

    False

    What is the definition of an empty relation in a set A?

    A relation R in a set A is called empty relation, if no element of A is related to any element of A, i.e., R = φ ⊂ A × A.

    ______ relation R in a set ______ is called universal relation, if each element of ______ is related to every element of _, i.e., R = ______ × ______.

    <p>A</p> Signup and view all the answers

    Which of the following relations is trivial?

    <p>Both</p> Signup and view all the answers

    The relation R = {(a, b) : a is sister of b} in the set A of all students of a boys school is the universal relation.

    <p>False</p> Signup and view all the answers

    Match the following relations with their definitions:

    <p>Empty relation = A relation R in a set A is called empty relation, if no element of A is related to any element of A. Universal relation = A relation R in a set A is called universal relation, if each element of A is related to every element of A. Trivial relation = Both the empty relation and the universal relation are sometimes called trivial relations.</p> Signup and view all the answers

    How many ways of representing a relation have been seen in Class XI?

    <p>2</p> Signup and view all the answers

    What is a relation R in a set A called if it is reflexive, symmetric, and transitive?

    <p>Equivalence relation</p> Signup and view all the answers

    A relation R in a set A is reflexive if (a, b) ∈ R, for every a ∈ A.

    <p>False</p> Signup and view all the answers

    What is the condition for a relation R in a set A to be symmetric?

    <p>If (a1, a2) ∈ R implies that (a2, a1) ∈ R, for all a1, a2 ∈ A.</p> Signup and view all the answers

    A relation R in a set A is called transitive if (a1, a2) ∈ R and (a2, a3) ∈ R implies that (a1, ____) ∈ R, for all a1, a2, a3 ∈ A.

    <p>a3</p> Signup and view all the answers

    In the relation R = {(T1, T2) : T1 is congruent to T2}, what can be said about the relation R?

    <p>R is an equivalence relation</p> Signup and view all the answers

    The relation R = {(L1, L2) : L1 is perpendicular to L2} is an equivalence relation.

    <p>False</p> Signup and view all the answers

    Match the following types of relations with their definitions:

    <p>Reflexive relation = If (a, a) ∈ R, for every a ∈ A. Symmetric relation = If (a1, a2) ∈ R implies that (a2, a1) ∈ R, for all a1, a2 ∈ A. Transitive relation = If (a1, a2) ∈ R and (a2, a3) ∈ R implies that (a1, a3) ∈ R, for all a1, a2, a3 ∈ A.</p> Signup and view all the answers

    What is the notation for 'a is related to b' in a binary relation R?

    <p>a R b</p> Signup and view all the answers

    What is a necessary condition for a relation R to be an equivalence relation?

    <p>R is reflexive, symmetric and transitive</p> Signup and view all the answers

    The relation R = {(a, b) : a ≤ b2} is reflexive.

    <p>False</p> Signup and view all the answers

    What is the difference between a reflexive relation and a symmetric relation?

    <p>A reflexive relation R satisfies the condition (a, a) ∈ R, whereas a symmetric relation R satisfies the condition (a, b) ∈ R ⇒ (b, a) ∈ R.</p> Signup and view all the answers

    A relation R is said to be transitive if (a, b) ∈ R and (b, c) ∈ R, then ______________________.

    <p>(a, c) ∈ R</p> Signup and view all the answers

    Which of the following relations is an equivalence relation?

    <p>R = {(a, b) : a and b are both odd or even}</p> Signup and view all the answers

    The relation R = {(x, y) : x – y is an integer} is transitive.

    <p>True</p> Signup and view all the answers

    Match the following relations with their properties:

    <p>R = {(a, b) : a ≤ b2} = Not reflexive, symmetric or transitive R = {(x, y) : y = x + 5 and x &lt; 4} = Not reflexive, symmetric or transitive R = {(x, y) : y is divisible by x} = Not reflexive or symmetric R = {(a, b) : a and b are both odd or even} = Reflexive, symmetric and transitive</p> Signup and view all the answers

    What is the difference between a binary relation and an equivalence relation?

    <p>A binary relation is a relation between two elements, whereas an equivalence relation is a special type of binary relation that satisfies the properties of reflexivity, symmetry and transitivity.</p> Signup and view all the answers

    Study Notes

    Types of Relations

    • A relation in a set A is a subset of A × A.
    • The empty set φ and A × A are two extreme relations.
    • Definition 1: A relation R in a set A is called an empty relation if no element of A is related to any element of A, i.e., R = φ ⊂ A × A.
    • Definition 2: A relation R in a set A is called a universal relation if each element of A is related to every element of A, i.e., R = A × A.

    Representing Relations

    • Relations can be represented using the raster method and set builder method.
    • Relations can also be expressed using the notation a R b if and only if b = a + 1.

    Properties of Relations

    • Definition 3: A relation R in a set A is called:
    • Reflexive, if (a, a) ∈ R, for every a ∈ A.
    • Symmetric, if (a1, a2) ∈ R implies that (a2, a1) ∈ R, for all a1, a2 ∈ A.
    • Transitive, if (a1, a2) ∈ R and (a2, a3) ∈ R implies that (a1, a3) ∈ R, for all a1, a2, a3 ∈ A.

    Equivalence Relations

    • Definition 4: A relation R in a set A is said to be an equivalence relation if R is reflexive, symmetric, and transitive.
    • Example: The relation R in the set of all triangles in a plane, given by R = {(T1, T2) : T1 is congruent to T2}, is an equivalence relation.

    Examples and Exercises

    • Example: The relation R in the set of all students of a boys school, given by R = {(a, b) : a is sister of b}, is the empty relation, and R′ = {(a, b) : the difference between heights of a and b is less than 3 meters} is the universal relation.
    • Exercise: Determine whether each of the following relations are reflexive, symmetric, and transitive.

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    Related Documents

    lemh101.pdf

    Description

    This chapter covers different types of relations, composition of functions, invertible functions, and binary operations. Learn about the properties and examples of relations and functions.

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