Which set theory axioms does the cyclic structure a ∈ b, b ∈ a satisfy?

[cragar]

Homework Statement

For each structure, draw a directed graph representing the membership relation. Then determine which of the following axioms is satisfied by the structure: Extensionality, Foundation, Pairing, Union
U= {a,b} a in b , and b in a

The Attempt at a Solution

The directed graph would be an arrow from a to b and then an arrow from b to a.
This fails foundation because a can't be in b and b can't be in a.
It satisfies Extensionality because a is a member of b and vice versa.
it satisfies pairing and union

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[andrewkirk]

I agree with your conclusions, but your justification of them is very terse and doesn't make it clear how the axioms that fail do so. Hence it may not get full marks from the lecturer. I would suggest writing out the justifications more fully, in symbolic form rather than prose. In particular, justifying that Pairing is not violated takes several steps, as you need to show that the set {a} is part of the structure.