7 8 Prelim Ma

PRELIMINARY
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.
There exists x, such that for any y (y is the intersection set), z is an element of y if and only if z is an element of every w and w is an element of x. That is written unofficially as:
∀x∃y∀z(z∈y↔∃w(∀w(z∈w)∧(w∈x))
I can't seem to find a way to make x, and y free and still write the statement in unofficial wff notation.

b.

If X is the empty set we look inside X and see that there is nothing. There are no W within X, because X is empty. So there can be no common elements among the W within X that can be made into a new set, the intersection set, and the intersection set fails to exist. Perhaps the definition for intersection could be amended to any X not equal to the empty set.

c.

The intersection of N is the elements that are common to all the sets contained within N. Start with a set at an arbitrary position, Q, within N. This Qth set is a set composed of every set prior to it, by definition. So the set before is smaller and has Q-1 elements in common. If we induct on length all the way back to the initial case we see that the only common element to all the sets from the initial to the Qth element is the initial element set itself. This initial element set is the null set, so therefore the null set is the intersection of N.

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