Mathematics Chapter 1: Relations and Functions
24 Questions
0 Views

Choose a study mode

Play Quiz
Study Flashcards
Spaced Repetition
Chat to lesson

Podcast

Play an AI-generated podcast conversation about this lesson

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.

    Studying That Suits You

    Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

    Quiz Team

    Related Documents

    Relations and Functions 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.

    More Like This

    Relations and Functions in Math
    5 questions

    Relations and Functions in Math

    ConciliatoryPythagoras avatar
    ConciliatoryPythagoras
    Set Theory and Relations Quiz
    10 questions

    Set Theory and Relations Quiz

    ProminentMaracas avatar
    ProminentMaracas
    BCA Math Unit 2: Set Theory and Functions
    8 questions

    BCA Math Unit 2: Set Theory and Functions

    DiplomaticMarsh7457 avatar
    DiplomaticMarsh7457
    Relations & Functions Concepts
    10 questions

    Relations & Functions Concepts

    EndorsedStatistics avatar
    EndorsedStatistics
    Use Quizgecko on...
    Browser
    Open
    Browser
    Browser