Relations and Functions Test

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Relations and Functions Test - 1

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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

Explanation:

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

B
none of these

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

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

Solution

Explanation:

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 x R y if x + 2 y = 8. The domain of R is

A
{2, 4, 6, 8}

B
{2, 4, 8}

C
{2, 4, 6}

D
{1, 2, 3, 4}

Solution

Explanation:

As x R y 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
x R y x – y is an even integer

B
x R y x + y is an even integer

C
x R y x = y

D
x R y x ≤ y

Solution

Explanation:

If R is a relation defined by xRy:ifx ⩽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
reflexive

C
anti – symmetric

D
symmetric

Solution

Explanation:

Any relation R is reflexive if x R x 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
Reflexive

B
symmetric

C
anti symmetric

D
not antisymmetric

Solution

Explanation:

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), (1, 2), (2, 1)}

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

C
none of these.

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

Solution

Explanation:

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
symmetric

B
none of these.

C
reflexive

D
transitive

Solution

Explanation:

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
symmetric

C
none of these.

D
reflexive

Solution

Explanation:

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
n²

C
2x2

D
2^_n2

Solution

Explanation:

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 =2^_n2

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Relations and Functions Test - 1

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Question 1

R = A ∩ B

R ⊂ A × B.

R = A × B

R = A ∪ B

Solution

Question 2

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

none of these

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

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

Solution

Question 3

{2, 4, 6, 8}

{2, 4, 8}

{2, 4, 6}

{1, 2, 3, 4}

Solution

Question 4

x R y x – y is an even integer

x R y x + y is an even integer

x R y x = y

x R y x ≤ y

Solution

Question 5

Transitive

reflexive

anti – symmetric

symmetric

Solution

Question 6

Reflexive

symmetric

anti symmetric

not antisymmetric

Solution

Question 7

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

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

none of these.

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

Solution

Question 8

symmetric

none of these.

reflexive

transitive

Solution

Question 9

transitive

symmetric

none of these.

reflexive

Solution

Question 10

2n

n2

2x2

2n2

Solution

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n²

2^_n2