Question

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

Answer 1

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.