Set Difference is Anticommutative

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 Equality of Sets:
 * $$S = T \iff \left({S \subseteq T}\right) \and \left({T \subseteq S}\right)$$

Also see

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