Set Difference is Anticommutative

Theorem
Set difference is an anticommutative operation:


 * $S = T \iff S \setminus T = T \setminus S = \O$

Proof
From Set Difference with Superset is Empty Set‎ we have:
 * $S \subseteq T \iff S \setminus T = \O$
 * $T \subseteq S \iff T \setminus S = \O$

The result follows from definition of set equality:
 * $S = T \iff \paren {S \subseteq T} \land \paren {T \subseteq S}$

Also see

 * Union is Commutative
 * Intersection is Commutative
 * Symmetric Difference is Commutative