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I noticed that XOR ($\oplus$) has a somewhat "mutualistic" property:

$\left(\left(A \oplus B\right)\iff C\right) \iff \left(A \iff \left(B \oplus C\right)\right)$

This can be easily checked via a truth table.

Saying that if C is equivalent with the value of A XOR B, then also A is equivalent with the value of B XOR C (and B is equivalent with the value of A XOR C).

Is there a name for this property?

I first thought it might be derived from the associative property, but then I checked that AND and OR (which have the associative property too) don't have this property.

DrFaulig
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2 Answers2

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Your observation is essentially that $B\oplus(A\oplus B)\iff A$; this follows from the fact that $\oplus$ is commutative and satisfies $X\oplus X=\bot$ for all $X$, where $\bot$ is FALSE. It’s the exact analogue of the fact that if $\mathbin\triangle$ is symmetric difference, then $B\mathbin\triangle(A\mathbin\triangle B)=A$, which follows from the commutativity of $\mathbin\triangle$ and the fact that $X\mathbin\triangle X=\varnothing$ for all $X$.

Here’s a slightly different way to look at it. For fixed $A$ (of the appropriate kind) the maps $X\mapsto A\oplus X$ and $X\mapsto A\mathbin\triangle X$ are involutions: apply either map twice, and you get back the original input. That is, $A\oplus(A\oplus X)=X$, and $A\mathbin\triangle(A\mathbin\triangle X)=X$. Your observation follows from the fact that this is true for all $A$ and the commutativity of $\oplus$.

Brian M. Scott
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From what I've read, one usually says that $\;\oplus\;$ and $\;\iff\;$ are mutually associative.

And since one is the negation of the other, in other words $$ A \oplus B \;\iff\; \lnot (A \iff B) $$ therefore I much prefer to use the symbols $\;\not\equiv\;$ and $\;\equiv\;$ instead: that makes the relationship between these two symbols clearly visible.

Then one can, for example, define the symmetric difference of sets $\;S\;$ and $\;T\;$ as $$ x \in S \triangle T \;\equiv\; x \in S \not\equiv x \in T $$ for every $\;x\;$, without the need for any parentheses because of the mutual associativity of $\;\equiv\;$ and $\;\not\equiv\;$.

MarnixKlooster ReinstateMonica
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