Question

Consider the relation \(R=\\{(a, a),(b, b),(c, c),(d, d),(a, b),(b, a)\\}\) on set \(A=\\{a, b, c, d\\}\) Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why.

Short answer

The given relation R is reflexive and symmetric, but not transitive.

Step-by-step solution

Check for reflexivity

In given relation R every element of set A is related to itself, i.e, (a, a), (b, b), (c, c), and (d, d) are in relation R. Therefore, the relation R is reflexive.

Check for symmetry

From the relation R, (a, b) and (b, a) both exist. There is no other ordered pairs in R, hence the condition for symmetry is also satisfied. Thus, the relation is symmetric.

Check for transitivity

For transitivity, we have to check if a third element exists where a is related to b and b is related to c then a must be related to c. Here, (a, b) and (b, a) are in the relation but (a, a) or (b, b) are not. Therefore, the property of transitivity doesn't hold, as there is no third pair to satisfy the condition.