Class XI Mathematics: Types of Relations and Functions

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Questions and Answers

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?

<p>{(a, b) : |a - b| ≥ 0} (C)</p> Signup and view all the answers

Why is the relation R in Example 3 not reflexive?

<p>A line cannot be perpendicular to itself (A)</p> Signup and view all the answers

What does a universal relation in a set A indicate?

<p>Each element is related to every other element (D)</p> Signup and view all the answers

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

<p>If L1 is perpendicular to L2, then L2 is always perpendicular to L1 (C)</p> Signup and view all the answers

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

<p>Empty relation (A)</p> Signup and view all the answers

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

<p>Universal relation (D)</p> Signup and view all the answers

Why is the relation R in Example 3 not transitive?

<p>Perpendicularity is not a transitive relation among lines (C)</p> Signup and view all the answers

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

<p>Each element is related to all other elements (B)</p> Signup and view all the answers

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

<p>{1, 2, 3} includes elements that relate to themselves (B)</p> Signup and view all the answers

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

<p>Symmetry (A)</p> Signup and view all the answers

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

<p>It is reflexive, symmetric, and transitive (B)</p> Signup and view all the answers

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

<p>(a – b) + (b – c) is divisible by 2 (A)</p> Signup and view all the answers

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

<p>All even integers (C)</p> Signup and view all the answers

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

<p>The equivalence class containing zero (A)</p> Signup and view all the answers

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

<p>Because E contains all even integers and O contains all odd integers (A)</p> Signup and view all the answers

Which of the following best defines an onto function?

<p>Range of the function is equal to the codomain (B)</p> Signup and view all the answers

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

<p>Because there are no two students with the same roll number (B)</p> Signup and view all the answers

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

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

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

<p>$f: R \to R$ given by $f(x) = 2x$ (D)</p> Signup and view all the answers

What is the defining characteristic of a bijective function?

<p>It is both one-one and onto (C)</p> Signup and view all the answers

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

<p>$f$ maps every real number to its double value (B)</p> Signup and view all the answers

Which of the following functions is both injective and surjective?

<p>f : Z → Z given by f (x) = x^2 (C)</p> Signup and view all the answers

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

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

Which function is shown to be one-one?

<p>f : R → R defined by f (x) = 3 – 4x (C)</p> Signup and view all the answers

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

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

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

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

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

<p>f : R → R given by f (x) = [x] (B)</p> Signup and view all the answers

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