Set Difference is Anticommutative
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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:
- $S = T \iff \paren {S \subseteq T} \land \paren {T \subseteq S}$
$\blacksquare$
Also see
Sources
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 1.1$: Theorem $1.7$
- 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): Exercise $1.1: \ 7$