Question

Let \(A=\\{a, b, c, d\\} .\) Suppose \(R\) is the relation $$ \begin{aligned} R=&\\{(a, a),(b, b),(c, c),(d, d),(a, b),(b, a),(a, c),(c, a)\\\ &(a, d),(d, a),(b, c),(c, b),(b, d),(d, b),(c, d),(d, c)\\} \end{aligned} $$ Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why.

Short answer

The relation R is reflexive, symmetric and transitive.

Step-by-step solution

Reflexive Property

Reflexive property asserts that every element of set A is related to itself. Here, set A = {a, b, c, d}. It is noted that (a, a), (b, b), (c, c) and (d, d) are elements of set R. Thus, set R is reflexive because every element of A is related to itself.

Symmetric Property

For the symmetric property to hold, every time (a, b) is in the set, so also must (b, a) be in the set. Checking the pairs in R, it is seen that the symmetric property also holds because every time (a, b) is part of the set, so is (b, a).

Transitive Property

The relation R is transitive if whenever (a, b) and (b, c) are in the set, (a, c) is also in R. Checking for transitivity, it is seen that for every pair of elements (a, b) and (b, c) in R, the pair (a, c) is also in R. Thus, R is transitive.