Question

Let a 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​

Answer 1

FUNCTIONS

Letbe the relation on the setgiven by. Then,is reflexive and symmetric but not transitive (b)is reflexive and transitive but not symmetric (c)is symmetric and transitive but not reflexive (d)is an equivalence relation

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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