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Syrus
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Homework Statement
Suppose A is a set and F ⊆ P(A). Let R = {(a,b) ∈ A x A | for every X ⊆ A\{a,b}, if X∪{a} ∈ F, then X∪{b} ∈ F}. Show that R is transitive.
* P(A) is the powerset of A.
Homework Equations
I am looking for an informal, "naive" proof- the text this exercise comes from is not axiomatic one.
The Attempt at a Solution
So far I have come up with:
Let a,b,c ∈ A and suppose (a,b) ∈ R and (b,c) ∈ R. Let X ⊆ A\{a,c} and also assume X∪{a} ∈ F. But then X∪{a} ∈ P(A), which in turn means X∪{a} ⊆ A.
*I am struggling with a way to show X∪{c} ∈ F, which is (at least what seems to me to be) what is required to prove (a,c) ∈ R.
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