Questions 7-8 Pre-Write

7. Write an official WFF expressing the Empty Set Axiom

Unofficial WFF:

∃y(∀z zЄy)

There exists a y such that for all z, z is not an element of y.

Possible official WFF:

∃y∀z¬(z Є y)

8. Intersection WFF

Statement: Given a set, there is a set ∩x consisting of all elements that belong to x.

∩x = ∀z(zЄy ↔ z belongs to every w in x)

∀z(zЄy ↔ ∀w(wЄx → zЄw)

Suppose that x=∅. For the formula above, inputting the empty set for x gives the result ∩∅=(zЄy), which is obviously incorrect. To account for this, we can rewrite the formula as:

∩x = ∀z(zЄy ↔ (x≠∅) ∧ (∀w(wЄx → zЄw)))

Similarly, ∩ N returns the empty set. This is incorrect as N itself is not empty, only its elements are.

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