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## Main Question or Discussion Point

My book presents the theorem that the empty set is a subset of every set as follows:

Let A be a set.

Let x be any object.

Assume x belongs to empty set.

Then x belongs to A.

Thus the empty set is a subset of A.

In other words, the proposition [(x belongs to empty set) implies (x belongs to A)] is true because the antecedent (x belongs to empty set) is always false, and a false statement may imply anything. But I really don't like this proof and I think it contradicts what my book previously stated: "We must be careful about making assumptions, because we can only be certain that what we proved will be true

But clearly the assumption that x belongs to empty set is always false.

I mean if you take this to be a valid proof, then I could just as well say that

[(x belongs to empty set) implies (empty set is not subset of A)]

thereby proving that the empty set is not a subset of any set...

So my question is, is a proof absolutely correct even if it is based on false assumptions?

I can see how to prove this theorem by contradiction: if it is not the case that [(x belongs to empty set) implies (x belongs to A)], then

[(x belongs to empty set) and (x doesnt belong to A)] is true.

But x doesn't belong to empty set. So the above is false, which is contradiction, and so empty set is subset of A.

But i'm just not convinced that the book's proof is valid...

Let A be a set.

Let x be any object.

Assume x belongs to empty set.

Then x belongs to A.

Thus the empty set is a subset of A.

In other words, the proposition [(x belongs to empty set) implies (x belongs to A)] is true because the antecedent (x belongs to empty set) is always false, and a false statement may imply anything. But I really don't like this proof and I think it contradicts what my book previously stated: "We must be careful about making assumptions, because we can only be certain that what we proved will be true

*when all the assumptions are true*."But clearly the assumption that x belongs to empty set is always false.

I mean if you take this to be a valid proof, then I could just as well say that

[(x belongs to empty set) implies (empty set is not subset of A)]

thereby proving that the empty set is not a subset of any set...

So my question is, is a proof absolutely correct even if it is based on false assumptions?

I can see how to prove this theorem by contradiction: if it is not the case that [(x belongs to empty set) implies (x belongs to A)], then

[(x belongs to empty set) and (x doesnt belong to A)] is true.

But x doesn't belong to empty set. So the above is false, which is contradiction, and so empty set is subset of A.

But i'm just not convinced that the book's proof is valid...