Relations and Functions Test

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Relations and Functions Test - 9

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Weekly Quiz Competition

Question 1
1 / -0

A relation R in a set A is called universal relation, if

A
every element of A is related to one element of A

B
no element of A is related to any element of A

C
each element of A is related to every element of A

D
one element of A is related to all elements of A

Solution

The relation R = A x A is called Universal relation.
Question 2
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A relation R in a set A is called reflexive,

A
if (b, b) ∈ R, for every a∈ A

B
if (a, b) ∈ R, for every a, b∈ A

C
if (a, a) ∈ R, for every a ∈ A

D
if (b, a) ∈ R, for every a, b ∈ A

Solution

A relation R on a non empty set A is said to be reflexive if x Rx for all x ∈ R , Therefore , R is reflexive.
Question 3
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A relation R on a non-empty set A is said to be an equivalence relation if

A
R is symmetric and transitive

B
R is reflexive, symmetric and transitive

C
R is reflexive and symmetric

D
R is reflexive and transitive

Solution

A relation R on a non empty set A is said to be reflexive if xRx for all x ∈ A . .
A relation R on a non empty set A is said to be symmetric if xRy ⇔ yRx, for all x , y ∈A .
A relation R on a non empty set A is said to be transitive if xRy and yRz ⇒ x Rz, for all x,y,z ∈ A.
An equivalence relation satisfies all these three properties.
Question 4
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Let A be the set of all students of a boys school. The relation R on A is given by R = {(a, b) : a is sister of b} is

A
Reflexive

B
Universal

C
Trivial

D
Empty

Solution

Because, the relation is defined over the set A be the set of all students of a boys school.
Question 5
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Let L be the set of all lines in a plane and R be the relation on L defined as R = {(L₁, L₂): L₁ is perpendicular to L₂}. Then R is

A
symmetric and transitive but not Reflexive

B
Reflexive and transitive but not symmetric

C
Symmetric but neither reflexive nor transitive.

D
Symmetric and reflexive but not transitive

Solution

The relation R is symmetric only , because if L₁ is perpendicular to L_{2 ,}then L₂ is also perpendicular to L₁,but no other cases that is reflexive and transitive is not possible.
Question 6
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The relation R on the set Z of integers given by R = {(a, b): 2 divides a – b} ,∀ a, b ∈ Z is

A
a universal relation

B
an empty relation

C
an equivalence relation

D
non commutative relation
Solution
1. Since a – a = 0 , and 0 is divisible by 2 , therefore, aRa i.e. R is reflexive.
2. If aRb , then a – b is divisible by 2. ⇒ - ( a- b ) is divisible by 2. ⇒(b – a ) is divisible by 2. ⇒bRa i.e. R is symmetric. .
3. .If aRb and bRc , then a – b is divisible by 2 and b – c is divisible by 2 ⇒ a – b = 2q and b – c = 2q’ where q and q’ are integers. ⇒( a – b ) + ( b – c ) = 2 ( q + q’) ⇒ a – c =2( q + q’), but (q +q’) is an integer. ⇒(a –c ) is divisible by 2. ⇒aRc i.e. R is transitive. .
Question 7
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Let R be the relation defined in the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b) : both a and b are either odd or even}. Then R is

A
a universal relation

B
an empty relation

C
an equivalence relation

D
non commutative relation
Solution
Consider any a ,b , c ∈A .
1. Since both a and a must be either even or odd, so (a , a) ∈R ⇒R is reflexive.
2. Let (a ,b) ∈ R ⇒ both a and b must be either even or odd, ⇒ both b and a must be either even or odd, ⇒ ( b ,a) ∈R .Thus , (a ,b) ∈R ⇒ ( b ,a) ∈R ⇒R is symmetric.
3. Let (a ,b) ∈R and (b ,c) ∈R ⇒ both a and b must be either even or odd, also ,both b and c must be either even or odd, ⇒ all elements a, b and c must be either even or odd, ⇒ ( a ,c) ∈ R . Thus , (a ,b) ∈ R ⇒ ( b ,c) ∈ R ⇒ (a ,c) ∈ R ⇒R is transitive.

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Re-Attempt Weekly Quiz Competition

Relations and Functions Test - 9

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Relations and Functions Test - 9

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SHARING IS CARING

Question 1

every element of A is related to one element of A

no element of A is related to any element of A

each element of A is related to every element of A

one element of A is related to all elements of A

Solution

Question 2

if (b, b) ∈ R, for every a∈ A

if (a, b) ∈ R, for every a, b∈ A

if (a, a) ∈ R, for every a ∈ A

if (b, a) ∈ R, for every a, b ∈ A

Solution

Question 3

R is symmetric and transitive

R is reflexive, symmetric and transitive

R is reflexive and symmetric

R is reflexive and transitive

Solution

Question 4

Reflexive

Universal

Trivial

Empty

Solution

Question 5

symmetric and transitive but not Reflexive

Reflexive and transitive but not symmetric

Symmetric but neither reflexive nor transitive.

Symmetric and reflexive but not transitive

Solution

Question 6

a universal relation

an empty relation

an equivalence relation

non commutative relation

Solution

Question 7

a universal relation

an empty relation

an equivalence relation

non commutative relation

Solution

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