7 8 Final Ma

FINAL

7. Writing out the statement in words we can say: there exists a set x, such that for any y that exists, y is not an element of x.
∃x∀y¬(y∈x).

8. a. Within the set X there are smaller sets. We are looking for all the elements that are common to all the smaller sets, W, that are within X. That is written unofficially as: ∀z((z∈y)↔∀w((w∈x)→(z∈w)).

b. The definition does not make sense if we have the empty set because the part of the expression on the right hand side that involves "(w∈x)" is vacuously true, since there are no sets 'w' contained within the empty set. Thus, we get the intersection being the set of all sets, which is rather counter-intuitive. If we amend our definition, adding that x≠∅ on the right hand side of the informal well-formed function in part a, we can prevent this case.

c. Take the set of natural numbers N to be X from our unofficial well-formed function in part a. Note that for any z, z is not an element of empty set, but the empty set is an element of X. So the right hand side of our unofficial well-formed function must evaluate to false for any z. So there is no z that is an element of Y. If Y, the intersection, contains no elements than it must be the null set. Thus, we have the intersection of N is the null set.