A relation R is said to be symmetric iff for any pair a and b if a is in the relation R with b, then also b is in the relation R with a.
\[R\quad is\quad symmetric \Leftrightarrow\forall a,b \in A, (aRb) \Rightarrow (bRa)\]
A Practical tip: A matrix that represent a symmetric relation should be symmetric in respect to its diagonal and it can (but not necessarily) have elements on the diagonal itself. For example, the following symmetrical relation
aRb, bRa
Would be represented in this way:
Add the least possible amount of crosses to the matrix below to transform the depicted relation into a symmetric relation on the set A = {a, b, c, d}
| a | b | c | d | |
|---|---|---|---|---|
| a | X | |||
| b | X | |||
| c | X | |||
| d | X |