The relationship between logical implication and the subset relation
[math]A\subseteq B[/math] (and we say "A is a subset of B") if and only if every element of [math]A[/math] also belongs to [math]B[/math]
That is: [math][A\subseteq B]\iff\forall x[x\in A\implies x\in B][/math]
We may often write:
- [ilmath]\forall x\in A[x\in B][/ilmath] instead.
This is easily seen to be equivalent as if [ilmath]A[/ilmath] is empty (so there is no [ilmath]x\in A[/ilmath] to speak of) the implication is semantically true, and the forall is vacuously true.
- Definition 3.10 (p10) - Introduction to Set Theory, Third Edition (Revised and Expanded) - Karel Hrbacek and Thomas Jech