What do you mean by reflexive, transitive, and symmetric relation? Explain with some suitable examples.

Solution:

A relation is a subset of a cross-product of two sets. Thus, if A and B are two sets, then a relation from A to B is a subset of A×B

Reflexive Property
According to this property, if (p, p) ∈ R, for every p ∈ N

For example, consider a relation R = {(P, Q): OP = OQ}.  Now (P, P) ∈ R since OP = OP for any point P. So, the relation is a reflexive relation.

Symmetric Property
According to this property if (p, q) ∈ R, then (q, p) ∈ R should also hold true.

Let's consider the same relation R. Again this relation is symmetric as if (P, Q) ∈ R ⇒ (Q, P) ∈ R

Transitive Property
According to this property, if (p, q), (q, r) ∈ R, then (p, r) ∈ R.

For example, this relation R is transitive as if (P, Q) ∈ R, (Q, S) ∈ R ⇒ (P, S) ∈ R

Since OP = OQ, OQ = OS ⇒ OP = OS for all P, Q, S.

If a relation is reflexive, transitive, and symmetric then it is an equivalence relation.

Thus, (i) A relation is reflexive if (a, a) ∈ R for every a ∈ A, (ii) A relation is transitive if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R, (iii) A relation is symmetric if (a, b) ∈ R then (b, a) ∈ R.

What do you mean by reflexive, transitive, and symmetric relation? Explain with some suitable examples.

Summary:

 A relation is reflexive if (a, a) ∈ R for every a ∈ A, (ii) A relation is transitive if (a, b) ∈ R and (b, c) ∈ R then (a, c) ∈ R, (iii) A relation is symmetric if (a, b) ∈ R then (b, a) ∈ R.