# Let F be a family of sets. Prove that if ∅ ∈ F, then ∩F = ∅.?

Relevance
• Proof 1.

If A is a subset of F, then ∩F is a subset of A

∅ is a subset of F, therefore ∩F is a subset of ∅.

Since ∅ is the only subset of ∅, ∩F = ∅.

Proof2.

Assume ∩F <> ∅. Then there is some element x such that x is in every set in F. In particular, x is in ∅. That's a contradiction. Therefore it can't be the case that ∩F <>∅, and so ∩F must equal ∅.

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• 1. The elements in ∩F are in all the sets of F, hence **∩F is a subset of every set in F**.

2. Since ∅ ∈ F and by 1. we have that **∩F is a subset of ∅**.

3. Also, **the empty set ∅ is a subset of every set** since there is no element in ∅ which doesn't belong to another set.

4. We have therefore just shown by 2. that **∩F is a subset of ∅** and by 3. that **∅ is a subset of ∩F**. Together these imply that ∩F=∅. ■

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• This is perfect nonsense. Of course the intersection of all the sets in F can be empty. For example, suppose F is a family of sets of real numbers, and suppose the sets in F are (a) the odd numbers and (b) the even numbers and (c) the empty set. The intersection of all the sets in F is empty. Your teacher is an idiot.

• Umm, I think you need to reread the question, which does say the intersection is empty.

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