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Is it possible to have a relation on the set {a, b, c} that is both symmetric and transitive but not reflexive

User Vishal Tarkar
by
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1 Answer

14 votes
14 votes

Answer:

Yes, it is possible to have a relation on the set {a, b, c} that is both symmetric and transitive but not reflexive

Explanation:

Let

Set A={a,b,c}

Now, define a relation R on set A is given by

R={(a,a),(a,b),(b,a),(b,b)}

For reflexive

A relation is called reflexive if (a,a)
\in R for every element a
\in A


(c,c)\\otin R

Therefore, the relation R is not reflexive.

For symmetric

If
(a,b)\in R then
(b,a)\in R

We have


(a,b)\in R and
(b,a)\in R

Hence, R is symmetric.

For transitive

If (a,b)
\in R and (b,c)
\in R then (a,c)
\in R

Here,


(a,a)\in R and
(a,b)\in R


\implies (a,b)\in R


(a,b)\in R and
(b,a)\in R


\implies (a,a)\in R

Therefore, R is transitive.

Yes, it is possible to have a relation on the set {a, b, c} that is both symmetric and transitive but not reflexive.

User Zasz
by
3.1k points
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