Question

R = {(A, B) | Both A & B live in the same city}
then R is??
​

Answer 1

Step-by-step explanation:

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12th

Maths

Relations and Functions

Introduction to Relations

Relation R in the set A of ...

MATHS

Relation R in the set A of human beings in a town at a particular time given by R={(x,y):xandyliveinthesamelocality}

enter 1-reflexive and transitive but not symmetric

2-reflexive only

3-Transitive only

4-Equivalence

5-None

MEDIUM

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ANSWER

R={(x,y):x and y live in the same locality}

Clearly (x,y)∈R as x and x is the same human being.

∴R is reflexive.

If (x,y)∈R, then x and y live in the same locality.

⇒y and x live in the same locality.

⇒(y,x)∈R

∴R is symmetric.

Now, let (x,y)∈R and (y,z)∈R.

⇒ x and y live in the same locality and y and z live in the same locality.

⇒ x and z live in the same locality.

⇒(x,z)∈R

∴R is transitive.

Hence, R is reflexive, symmetric and transitive.

Answer 2

SOLUTION

TO DETERMINE

Let relation R be defined as R={(A,B)|both A and B live in the same city}.Pick out the correct statement (s).

• R is symmetric.

• R is anti-symmetric.

• R is transitive.

• R is reflexive.

EVALUATION

Here the given relation R is defined by

R = { (A,B) | both A and B live in the same city }

CHECKING FOR REFLEXIVE

Clearly ( A, A) ∈ R

Since A and A always live in the same city

So (A, A) ∈ R

So R is Reflexive

CHECKING FOR SYMMETRIC

Let (A , B ) ∈ R

⇒A and B live in the same city

⇒B and A live in the same city

⇒(B , A) ∈ R

Thus (A , B ) ∈ R implies (B , A ) ∈ R

So R is symmetric

CHECKING FOR TRANSITIVE

Let (A , B ) ∈ R and (B , C ) ∈ R

⇒A , B live in the same city and B , C live in the same city

⇒ A and C live in the same city

⇒(A , C ) ∈ R

Thus (A , B ) ∈ R and (B , C ) ∈ R implies (A , C ) ∈ R

R is transitive

Hence R is an equivalence relation

CHECKING FOR ANTI SYMMETRIC

Let (A , B ) ∈ R and (B , A) ∈ R

⇒A , B live in the same city and B , A live in the same city

It does not mean A and B are same

So R is not anti-symmetric.

FINAL ANSWER

Hence the correct statements are

• R is symmetric

• R is transitive.

• R is reflexive

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