Is “The empty set is a subset of any set” a convention?

Recently I learned that for any set A, we have A.

I found some explanation of why it holds.

A means “for every object x, if x belongs to the empty set, then x also belongs to the set A”. This is a vacuous truth, because the antecedent (x belongs to the empty set) could never be true, so the conclusion always holds (x also belongs to the set A). So A holds.

What confused me was that, the following expression was also a vacuous truth.

For every object x, if x belongs to the empty set, then x doesn’t belong to the set A.

According to the definition of the vacuous truth, the conclusion (x doesn’t belong to the set A) holds, so would be true, too.

Which one is correct? Or is it just a convention to let \varnothing\subset A?

Answer

There’s no conflict: you’ve misinterpreted the second highlighted statement. What it actually says is that \varnothing and A have no element in common, i.e., that \varnothing\cap A=\varnothing. This is not the same as saying that \varnothing is not a subset of A, so it does not conflict with the fact that \varnothing\subseteq A.

To expand on that a little, the statement B\nsubseteq A does not say that if x\in B, then x\notin A; it says that there is at least one x\in B that is not in A. This is certainly not true if B=\varnothing.

Attribution
Source : Link , Question Author : Searene , Answer Author : Brian M. Scott

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