Xor

Xor is a logical binary operation which takes two values of the set of $\mathbb{Z}/2$ and applies the “exclusive or” function which only returns 1 if both operands are different and will otherwise return 0.

The operation is noted as $\otimes$ throughout my work. \otimes in LaTeX.

Arg 1	Arg 2	Return
$1$	$1$	$0$
$1$	$0$	$1$
$0$	$1$	$1$
$0$	$0$	$0$

Truth table of Xor Note that this is an inversion of $⟺$

Decomposition of Xor

see also Decomposition of Biconditional $x\otimes y$ = $\neg (x⟺y)$ = $(x\vee y)\wedge \neg (x\wedge y)$ = $(x\vee y)\wedge (\neg x\vee \neg y)$

Self-Cancellation Property

Xor is a fascinating operation because for any elements $a,b\in S$, $b\otimes a\otimes a=b$ This is because in the monoid $(\{0,1\},\otimes )$, you have that $e=0$ and $\forall a\in \{0,1\}:a\otimes a=0$

This principle abides by the Law requirements therefore any monoid in the Variety generated by $(\{0,1\},\otimes )$ exhibits this. This is why we see it in Binary Numbers at scale and in programming.