Set Difference is Anticommutative

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Theorem

Set difference is an anticommutative operation:

$S = T \iff S \setminus T = T \setminus S = \varnothing$


Proof

From Set Difference with Superset is Empty Set‎ we have:

$S \subseteq T \iff S \setminus T = \varnothing$
$T \subseteq S \iff T \setminus S = \varnothing$

The result follows from definition of set equality:

$S = T \iff \left({S \subseteq T}\right) \land \left({T \subseteq S}\right)$

$\blacksquare$


Also see


Sources