Prove the following statement is true or give a counterexamp

Prove the following statement is true, or give a counterexample: Every commutative binary operation on a set having just two elements is associative.

Solution

Let a and b be the elements

All we need is one example of a multiplication table where associativity doesn\'t hold. Here is one.
_ab
aba
baa

This says that aa = b, bb=a, ab=ba=a
It\'s symmetrical wrt the diagonal, so it\'s commutative.

a(ab) = aa = b
(aa)b = bb = a
They aren\'t equal, so this is not associative.

The statement is false.