# 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: $[A\subseteq B]\iff\forall x[x\in A\implies x\in B]$[1]

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

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.

## References

1. Definition 3.10 (p10) - Introduction to Set Theory, Third Edition (Revised and Expanded) - Karel Hrbacek and Thomas Jech