Question #161394

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



1
Expert's answer
2021-02-24T06:54:40-0500

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

Then xΘy=yΘxx\Theta y=y\Theta x (commutativity).

0Θ(1Θ1)=0Θ0=10\Theta(1\Theta 1) = 0\Theta 0=1

(0Θ1)Θ1=1Θ1=0(0\Theta1)\Theta 1 = 1\Theta 1=0

0Θ(1Θ1)(0Θ1)Θ10\Theta(1\Theta 1)\ne(0\Theta1)\Theta 1

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


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