Relations and Functions Test

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Relations and Functions Test - 6

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Weekly Quiz Competition

Question 1
1 / -0

If R is a relation from a non – empty set A to a non – empty set B, then

A
R = A × B

B
R ⊂ A × B

C
R = A ∪ B

D
R = A ∩ B

Solution

Let A and B be two sets. Then a relation R from set A to set B is a subset of A×B.Thus, R is a relation from A to B ⇔ R ⊆ A × B.
Question 2
1 / -0

Let A = { 2 , 3 , 6 }. Which of the following relations on A are reflexive ?

A
R₂ = {( 2,2 ) , ( 3,3 ) , ( 3,6 ) , ( 6,3 )}

B
R₃ = {( 2,2 ) , ( 3,6 ) , ( 2,6 )}

C
R₁ = {( 2,2 ) , ( 3,3 ) , ( 6,6 )}

D
None of these

Solution

R₁ is a reflexive on A, because ( a,a ) ∈ R₁for each a ∈ A
Question 3
1 / -0

Let R be the relation on N defined as xRy if x + 2 y = 8. The domain of R is

A
{2, 4, 6}

B
{2, 4, 8}

C
{2, 4, 6, 8}

D
{1, 2, 3, 4}

Solution

As xRy if x + 2 y = 8 , therefore, domain of the relation R is given by x = 8 – 2y ∈ N. When y = 1, ⇒ x = 6, when y = 2, ⇒ x =4 , when y =3 , ⇒ x = 2. Therefore, domain is { 2, 4, 6 }.
Question 4
1 / -0

Which of the following is not an equivalence relation on I, the set of integers: x, y

A
xRy, x – y is an even integer

B
xRy, x = y

C
xRy, x + y is an even integer

D
xRy, x ≤ y

Solution

If R is a relation defined by xRy: if x⩽y, then R is reflexive and transitive. But, it is not symmetric. Hence, R is not an equivalence relation.
Question 5
1 / -0

Let A = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c) } be a relation on A. Here, R is

A
transitive

B
symmetric

C
anti – symmetric

D
reflexive

Solution

Any relation R is reflexive if xRx for all x ∈ R. Here, (a, a), (b, b), (c, c)∈ R. Therefore, R is reflexive.
Question 6
1 / -0

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

A
not anti symmetric

B
symmetric

C
anti symmetric

D
Reflexive

Solution

A relation R on a non empty set A is said to be reflexive if xRx for all x ∈ R , Therefore , R is not reflexive.
A relation R on a non empty set A is said to be symmetric if xRy ⇔ 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 xRy and yRx ⇒ x = y , for all x , y ∈ R. Therefore, R is not antisymmetric.
Question 7
1 / -0

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 8
1 / -0

Let A = {1, 2, 3}, then the relation R = {(1, 1), (2, 2), (1, 3)} on A is

A
transitive

B
symmetric

C
reflexive

D
None of these.

Solution

The given relation is not reflexive, as (3,3)∉R, The given relation is not symmetric, as (1,3)∈ R, but (3,1) ∉R, The given relation is transitive as (1,1) )∈ R and (1,3) )∈ R.
Question 9
1 / -0

Let A = {1, 2, 3}, then the relation R = {(1, 1), (1, 2), (2, 1)} on A is

A
transitive

B
reflexive

C
symmetric

D
None of these.

Solution

A relation R on a non empty set A is said to be symmetric iff xRy ⇔ yRx, for all x , y ∈ R Clearly, (1, 2), and (2, 1) both lies in R. Therefore, R is symmetric.
Question 10
1 / -0

If A is a finite set containing n distinct elements, then the number of relations on A is equal to

A
2n

B
2×2

C
2ⁿ²

D
n²

Solution

The number of elements in A x A is n x n = n². Hence ,the number of relations on A = number of subsets of A x A = 2^nxn =2ⁿ².

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Relations and Functions Test - 6

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Relations and Functions Test - 6

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Question 1

R = A × B

R ⊂ A × B

R = A ∪ B

R = A ∩ B

Solution

Question 2

R2 = {( 2,2 ) , ( 3,3 ) , ( 3,6 ) , ( 6,3 )}

R3 = {( 2,2 ) , ( 3,6 ) , ( 2,6 )}

R1 = {( 2,2 ) , ( 3,3 ) , ( 6,6 )}

None of these

Solution

Question 3

{2, 4, 6}

{2, 4, 8}

{2, 4, 6, 8}

{1, 2, 3, 4}

Solution

Question 4

xRy, x – y is an even integer

xRy, x = y

xRy, x + y is an even integer

xRy, x ≤ y

Solution

Question 5

transitive

symmetric

anti – symmetric

reflexive

Solution

Question 6

not anti symmetric

symmetric

anti symmetric

Reflexive

Solution

Question 7

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

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

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

None of these

Solution

Question 8

transitive

symmetric

reflexive

None of these.

Solution

Question 9

transitive

reflexive

symmetric

None of these.

Solution

Question 10

2n

2×2

2n2

n2

Solution

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R₂ = {( 2,2 ) , ( 3,3 ) , ( 3,6 ) , ( 6,3 )}

R₃ = {( 2,2 ) , ( 3,6 ) , ( 2,6 )}

R₁ = {( 2,2 ) , ( 3,3 ) , ( 6,6 )}

2ⁿ²

n²