Which logical connective corresponds to the set relationship A ⊆ B?

 

The definition of complementation may seem to be unnecessarily complicated. Why didn’t we just define “~ B” to be {x: x ∉ B}? The problem is that {x: x ∉ B} is too large. For example, suppose that B = {2, 4, 6, 8}. Then {x: x ∉ B} contains all of the following (and more!):

 

It is quite reasonable that the integers 1, 3, 5, 7 should be included in “~B,” and, depending on the context, we might want to include the real numbers greater than 25 as well. But it is quite unlikely that we would want to include any of the other items. Certainly, knowing that my uncle Wilbur is not a member of the set B would contribute little to any discussion of B. As we have observed earlier in the chapter, mathematical concepts and proofs always occur within the context of some mathematical system. It is customary for the elements of the system to be called the universal set. Then any set under consideration is a subset of this universal set. If the universal set in a particular discussion were the integers Z, then the nonintegral real numbers greater than 25 would not be included in {x: x ∉ B}. On the other hand, if the universal set were taken to be R, then they would be included