Question

Let a relation R be defined as R = { ( A , B ) | Both A and B live in the same city}.

Answer 1

Dear what you want to find?

Relation? or Cartesian of R?

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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