# Are Disjoint Set Assertions Commutative?

If I were to say that A intersect B was the empty set, does that imply that B intersect A is also the empty set?

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How could this fail? (It is not a rhetorical question.) – Andrés Caicedo Jan 19 '11 at 23:58

Yes.

Because $A \cap B = \{ x | x \in A \text{ and } x \in B \}$ and surely "and" is commutative.

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A ∩ B := {(x ∊ U)|(x ∊ A) ⋀ (x ∊ B)} You can see that this is a logical statement of the form P ⋀ Q and any logic textbook will tell you that P ⋀ Q = Q ⋀ P, i.e. that ⋀ is commutative, that seems to me to be the justification.

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