The relationship between logical implication and the subset relation

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Definition

$$A\subseteq B$$ (and we say "A is a subset of B") if and only if every element of $$A$$ also belongs to $$B$$

That is:

$$[A\subseteq B]\iff\forall x[x\in A\implies x\in B]$$

Note: 16/1/2017 by Alec (talk) 17:36, 16 January 2017 (UTC)

We may often write:

• $\forall x\in A[x\in B]$ instead.

This is easily seen to be equivalent as if $A$ is empty (so there is no $x\in A$ to speak of) the implication is semantically true, and the forall is vacuously true.