Question

Suppose R is a symmetric relation on a set A. Is the complement
of R also symmetric?

Answer 1

The complement of a symmetric relation is also symmetric.

Step-by-step explanation:

A symmetric relation on a set A is a relation in which if (x, y) belongs to R, then (y, x) also belongs to R. The complement of a relation R is denoted as R' and consists of all the pairs (x, y) that do not belong to R.

To determine if the complement of R is symmetric, we need to check if for every pair (x, y) in R', the pair (y, x) also belongs to R'.

If (x, y) is not in R, then (y, x) definitely cannot be in R, since R is symmetric. Therefore, the complement of R is also symmetric.

Answer 2

The complement of a symmetric relation R on a set A is not necessarily symmetric because the absence of one pair does not guarantee the absence of its inverse pair in the complement.

Step-by-step explanation:

The student has asked whether the complement of a symmetric relation R on a set A is also symmetric. A symmetric relation is one where if (a, b) is in R, then (b, a) is also in R. For the complement of R, denoted as R', to be symmetric, we would need that if (a, b) is not in R, then (b, a) is also not in R. However, this is not necessarily the case because the absence of one pair does not imply the absence of the flipped pair in the complement. Therefore, we cannot conclude that the complement of a symmetric relation is symmetric.