Set theory representation of material implication

[friend]

Just checking here.

Propositional logic connectives like AND and OR have analogs or representations in set theory. For example, the logical connective AND is represented in set theory by intersection, an element of X AND Y is the element of the intersection of sets X and Y. And similarly, the logical connective of OR is represented in set theory by union, an element of X OR Y is the element of the union of sets X or Y.

So what is the set theoretical representation of material implication. When I googled "set theory of material implication", I did not get anything explicit. Maybe I'm not searching the right phrases. Anyway, is the set representation of implication simply the subset, an element is a member of a set X implied by Y if X is a subset of Y. Is this right, or am I missing something?

This is very important to me. I'd like to have a firm foundation for material implication in terms of set theory. Is this a well known concept? Are there any on-line references I can review and quote? Thanks.

 

[Dragonfall]

See http://en.wikipedia.org/wiki/Johnston_diagram

 

[HallsofIvy]

Yes, it's well known.

The set theory version of material implication is "subset". If P is the set of all things for which statement p is true and Q the set of all things for which statement q is true, then "If p then q" can be represented as "Q is a subset of P".