Final Solution 17 Maria Acevedo (Class 2.27)

17. Statement: For all sets x,y, it is impossible for both x∈y and y∈x to hold.

Proof: Suppose that there exists arbitrary sets x and y such that the x∈y and y∈x both hold. The set z is the set containing both x and y, z = {x, y}. By foundation, z must be disjoint from either x or y. Two sets are said to be disjoint sets if they have no element in common. However, x ∩ z ∋ y and y ∩ z ∋ x means that the set is not disjoint. Thus, the statement violates foundation and cannot hold.

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