30 Questions
What is a relation in a set A defined as?
A subset of A
Which property must a relation have to be considered an equivalence relation?
Reflexive, symmetric, and transitive
In Example 2, why is the relation R considered reflexive?
Every element in T is congruent to itself
Which example represents an empty relation in a set A?
{(a, b) : |a - b| ≥ 0}
Why is the relation R in Example 3 not reflexive?
A line cannot be perpendicular to itself
What does a universal relation in a set A indicate?
Each element is related to every other element
In Example 3, why is the relation R considered symmetric?
If L1 is perpendicular to L2, then L2 is always perpendicular to L1
Which term refers to relations where no elements are related to any other element?
Empty relation
What is the relation R = {(a, b) : |a - b| ≥ 0} named as?
Universal relation
Why is the relation R in Example 3 not transitive?
Perpendicularity is not a transitive relation among lines
In a universal relation in set A, what is the relationship between elements?
Each element is related to all other elements
In Example 4, why is the relation R considered reflexive?
{1, 2, 3} includes elements that relate to themselves
Which property does the relation R in the set Z of integers lack?
Symmetry
Why is it stated that the relation R in the set Z is an equivalence relation?
It is reflexive, symmetric, and transitive
What does the statement 'a – c = (a – b) + (b – c) is even' imply?
(a – b) + (b – c) is divisible by 2
Which integers are related to zero in the relation R in the set Z?
All even integers
What does the subset E consisting of all even integers represent?
The equivalence class containing zero
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
Which of the following best defines an onto function?
Range of the function is equal to the codomain
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
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 $f(x) = 2x = 1$
In the given examples, which function is both one-one and onto?
$f: R \to R$ given by $f(x) = 2x$
What is the defining characteristic of a bijective function?
It is both one-one and onto
Why is the function $f: R \to R$, given by $f(x) = 2x$, considered onto?
$f$ maps every real number to its double value
Which of the following functions is both injective and surjective?
f : Z → Z given by f (x) = x^2
Which function is proven to be neither one-one nor onto?
Greatest Integer Function f : R → R, given by f (x) = [x]
Which function is shown to be one-one?
f : R → R defined by f (x) = 3 – 4x
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)
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
Which function can be concluded as having a domain and codomain both in real numbers?
f : R → R given by f (x) = [x]
Explore the concept of relations and functions in Mathematics Class XI, including different types of relations, composition of functions, invertible functions, and binary operations. Learn how to identify and analyze these mathematical structures.
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