The relationship between logical implication and the subset relation

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Definition

AB

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

That is: [AB]x[xAxB]

[1]

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

We may often write:

  • xA[xB] instead.

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

References

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