Answer to Question #161394 in Abstract Algebra for k

Question

Prove or give a counter example: “Every commutative binary operation on a set having just 2 elements is associative.”

Answer 1

Let us consider a set {0,1} with two elements and an operation $\Theta$ which acts by the rule: $x\Theta y = 1-\min (x,y)$

Then $x\Theta y=y\Theta x$ (commutativity).

$0\Theta(1\Theta 1) = 0\Theta 0=1$

$(0\Theta1)\Theta 1 = 1\Theta 1=0$

$0\Theta(1\Theta 1)\ne(0\Theta1)\Theta 1$

Therefore, commutative operations on the sets with two elements are not necessary associative.