Class XI Mathematics: Types of Relations and Functions

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What is a relation in a set A defined as?

A subset of A (correct)

A superset of A

A superset of B

A subset of B

Which property must a relation have to be considered an equivalence relation?

Reflexive, asymmetric, and transitive

Reflexive, symmetric, and transitive (correct)

Reflexive, transitive, and antisymmetric

Symmetric, transitive, and antisymmetric

In Example 2, why is the relation R considered reflexive?

Every element in T is congruent to at least one other element

Every element in T is not congruent to any other element

Every element in T is congruent to itself (correct)

Every element in T is not congruent to itself

Which example represents an empty relation in a set A?

{(a, b) : |a - b| ≥ 0} Signup and view all the answers

Why is the relation R in Example 3 not reflexive?

A line cannot be perpendicular to itself Signup and view all the answers

What does a universal relation in a set A indicate?

Each element is related to every other element Signup and view all the answers

In Example 3, why is the relation R considered symmetric?

If L1 is perpendicular to L2, then L2 is always perpendicular to L1 Signup and view all the answers

Which term refers to relations where no elements are related to any other element?

Empty relation Signup and view all the answers

What is the relation R = {(a, b) : |a - b| ≥ 0} named as?

Universal relation Signup and view all the answers

Why is the relation R in Example 3 not transitive?

Perpendicularity is not a transitive relation among lines Signup and view all the answers

In a universal relation in set A, what is the relationship between elements?

Each element is related to all other elements Signup and view all the answers

In Example 4, why is the relation R considered reflexive?

{1, 2, 3} includes elements that relate to themselves Signup and view all the answers

Which property does the relation R in the set Z of integers lack?

Symmetry Signup and view all the answers

Why is it stated that the relation R in the set Z is an equivalence relation?

It is reflexive, symmetric, and transitive Signup and view all the answers

What does the statement 'a – c = (a – b) + (b – c) is even' imply?

(a – b) + (b – c) is divisible by 2 Signup and view all the answers

Which integers are related to zero in the relation R in the set Z?

All even integers Signup and view all the answers

What does the subset E consisting of all even integers represent?

The equivalence class containing zero Signup and view all the answers

Why are no elements of E related to elements of O in the set Z?

Because E contains all even integers and O contains all odd integers Signup and view all the answers

Which of the following best defines an onto function?

Range of the function is equal to the codomain Signup and view all the answers

In Example 7, why is the function $f: A \to N$ considered one-one?

Because there are no two students with the same roll number Signup and view all the answers

Why is the function $f: N \to N$, given by $f(x) = 2x$, considered not onto?

Because for 1 ∈ N, there does not exist any x in N such that $f(x) = 2x = 1$ Signup and view all the answers

In the given examples, which function is both one-one and onto?

$f: R \to R$ given by $f(x) = 2x$ Signup and view all the answers

What is the defining characteristic of a bijective function?

It is both one-one and onto Signup and view all the answers

Why is the function $f: R \to R$, given by $f(x) = 2x$, considered onto?

$f$ maps every real number to its double value Signup and view all the answers

Which of the following functions is both injective and surjective?

f : Z → Z given by f (x) = x^2 Signup and view all the answers

Which function is proven to be neither one-one nor onto?

Greatest Integer Function f : R → R, given by f (x) = [x] Signup and view all the answers

Which function is shown to be one-one?

f : R → R defined by f (x) = 3 – 4x Signup and view all the answers

In which case does a function from A to B exhibit a bijective relationship?

f : A × B → B × A such that f (a, b) = (b, a) Signup and view all the answers

Which function from N to N is proven to be neither injective nor surjective?

f : N → N given by f (n) = n if n is even, 2 if n is odd Signup and view all the answers

Which function can be concluded as having a domain and codomain both in real numbers?

f : R → R given by f (x) = [x] Signup and view all the answers