Sets, Relations and Functions Test

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Sets, Relations and Functions Test - 6

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Question 1
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R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)} be a relation on A = {1, 2, 3}, then R is

A
Symmetric

B
Anti symmetric

C
Not antisymmetric

D
Reflexive

Solution

A relation R on a non empty set A is said to be reflexive if fx Rx for all x ∈ R, Therefore , R is not reflexive.
A relation R on a non empty set A is said to be symmetric if fx Ry ⇔yRx, for all x, y ∈R Therefore, R is not symmetric.
A relation R on a non empty set A is said to be antisymmetric if fx Ry and y Rx ⇒x = y, for all x, y ∈R. Therefore, R is not antisymmetric.
Question 2
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If R is a relation from a set A to a set B and S is a relation from B to C, then the relation S ^∘ R.

A
Is from A to C

B
Is from C to A

C
Does not exist

D
None of these

Solution

If R is a relation from A →B and S is a relation fromB →C, then S ^∘ R is a composite function from A to C.
Question 3
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Let A be the set of all students of a boys school. The relation R in A given by R = {(a, b) : a is sister of b} is

A
Universal

B
Trivial

C
Empty

D
Reflexive

Solution

Because, the relation is defined over the set A which is the set of all students of a boys school.
Question 4
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The domain of definition of the function

A
(−∞,0]

B
[0,∞)

C
(1,∞)

D
None of these.

Solution

y is defined If -x ≥ 0, i.e.
Question 5
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A
The greatest integer function

B
A constant function.

C
The absolute value function

D
The signum function

Solution

By definition, The Signum function =
Question 6
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Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A ?

A
{(1, 1), (2, 2), (3, 3)}

B
{(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)}

C
{(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)}

D
None of these.

Solution

A relation R on a non empty set A is said to be reflexive iff xRx for all x ∈ R . A relation R on a non empty set A is said to be symmetric iff xRy ⇔yRx, for all x , y ∈R .
A relation R on a non empty set A is said to be transitive iff xRy and yRz ⇒xRz, for all x ∈ R. An equivalence relation satisfies all these three properties.
None of the given relations satisfies all three properties of equivalence relation.
Question 7
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The binary operation * defined on the set of integers as a ∗b = |a −b|-1 is:

A
Commutative

B
Associative

C
Not commutative

D
None of these.

Solution

Here * is commutative as b*a = |b −a|−1 = |a −b|−1 = a ∗b.
Because ,|−x| = |x| for all x ∈R.
Question 8
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Let T be the set of all triangles in a plane with R a relation in T given by R = {(T₁ , T₂ ) : T₁ is congruent to T₂ }. Then R is

A
Non commutative relation

B
A universal relation

C
An equivalence relation

D
An empty relation

Solution

Let T be the set of all triangles in a plane with R a relation in T given by R = {(T₁ , T₂ ): T₁ is congruent to T₂ }. (T₁ T₂ ) ∈ R iff T₁ is congruent to T₂ .
Reflexivity :T₁ ≅T₁ ⇒(T₁ T₁ ) ∈R . Symmetry :(T₁ , T₂ )∈R ⇒T₁ ≅T₂ ⇒T₂ ≅T₁ ⇒(T₂ , T₁ ) ∈R
Transitivity :(T₁ ,T₁ ) ∈R and (T₂ , T₃ ) ∈R ⇒T₁ ≅T₂ and T₂ ≅T₃ ⇒T₁ ≅T₃ ⇒(T₁ , T₃ ) ∈R .
Therefore, R is an equivalence relation on T.
Question 9
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The range of the function

A
{–1, 1}

B
{–1, 0, 1}

C
[–1, 1]

D
None of these.

Solution

The given function is defined as Signum function. i.e.

domain is R and Range is {-1 , 0 , 1}.
Question 10
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The function f (x) = x² + sin x is

A
Neither even nor odd

B
An odd function

C
An even function

D
A constant function.

Solution

For even function : f(-x) = f(x) , therefore, f(- x) = (− x)² + sin (− x)
= x² −sin x ≠ f(x).
For an odd function : f(-x) = - f(x) , therefore, f(-x) = (− x)² + sin (− x)
= x² − sin x
≠− f(x).
Therefore f(x) is neither even nor odd.
Question 11
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If A is a finite set containing n distinct elements, then the number of relations on A is equal to

A

B
2n

C
n²

D
None of these

Solution
Question 12
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R is a relation from { 11, 12, 13} to {8, 10, 12} defined by y = x –3. The relation R⁻¹

A
{(8, 11), (9, 12), (10, 13)}

B
{(11, 8), (13, 10)}

C
{(8, 11), (10, 13)}

D
None of these

Solution

R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x –3. The relation is given by x = y + 3,from{8, 10, 12} to {11, 12, 13} ⇒ relation = {(8,11),(10,13)}.
Question 13
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If A = {(1, 2, 3}, then the relation R = {(2, 3)} in A is

A
Symmetric and transitive only

B
Symmetric only

C
Transitive only

D
Not transitive

Solution

A = {1,2,3}
B = {(2,3)} is not reflexive or symmetric on A but it is transitive
∵if (a,b) exists but (b,c) does not exist then (a,c) does not need to exist and the relation is still transitive.
Question 14
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The range of the function f(x) = [sin x] is

A
[1, 1].

B
(–1, 1)

C
{–1, 0, 1}

D
{–1, 1}

Solution

The only possible integral values of sin x are {-1 ,0, 1}.
Question 15
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The period of the function f(x) = sin² x + tan x is

A
2 π

B
3 π

C
π

D
None of these

Solution

We have the function f(x) = sin² x + tan x, therefore, f(π+x)
= {sin² (π+x)+tan(π+x)}
= {sin² x+tanx} = f(x).
This implies that period of the function f(x) is π.