-1
$\begingroup$

Possible Duplicate:
Number of relations that are both symmetric and reflexive

Let $X$ be a set with $8$ elements.

How many binary relations on $X$ are either reflexive or symmetric or both?

show work. you need not simplify the answer.

  • 0
    It is the number which are reflexive *plus* the number which are symmetric *minus* the number which are both (to avoid double counting)2012-01-10

1 Answers 1

1

In a set with 8 elements, a binary relation, $R$ can be thought of as a set of pairs of elements of the set for which that relation is true. That is, $(a,b) \in R \leftrightarrow aRb$ is true. As such, if there are 8 elements, then there 64 possible pairs (order matters). If the relation is reflexive, then this implies that $(a,a) \in R$ for all $a$. So we know that 8 of the pairs must be in the relation. This leaves 56 pairs which can either be in or not be in. So there are $2^{56}$ possible reflexive relations.

Similar combinatorics can be applied to find the answer for symmetric and both.

  • 0
    Oh, yeah, you are completely correct. How dumb of me. Fixed in most recent edit.2012-01-10