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Set theory concern

For a given set, let's say \(A\), is it true that \(\varnothing \subset A \) ?

My personal thought is that it's not true, Since:

Let \(A = \{1,2,3,4 \}\), then to say that \(\varnothing \subset A\), we say that every element of \( \varnothing\) (nothing) also belongs to the set \(A\), which is strange to say since \(A = \{1,2,3,4\}\).

On the other hand to say that \(\varnothing \subset A\), nothing seems to change in the set \(A\) since \(\varnothing\) has no elements...

Would it be necessary to set a condition such that \(A = \varnothing\) ?

What am I missing ? Any thoughts ?

Note by Andrew Hucek
1 month ago

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It is true that \( \emptyset \subset A \) for all sets.

It is not true that "\( \emptyset\) is nothing". It is the empty set.

See Sets - Subsets.

Calvin Lin Staff - 1 month ago

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"Nothing" wasn't to the set, but the elements of it, I will correct it

Andrew Hucek - 1 month ago

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Great, so now you have the right statement. Since there are no elements in the set, it is true that all elements of the empty set are in A.

The empty set can sometimes be strange. A lot of statements about it are true, even seemingly contradictory ones. For example, "All the mini coopers I own are red" and "All the mini coopers I own are blue" are both true statements, because the empty set (mini coopers I own) belong to every other set out there.

Calvin Lin Staff - 1 month ago

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@Calvin Lin The empty set is sometimes confusing, one has to be more careful how to use the english in mathematics the correct way.

Andrew Hucek - 1 month ago

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The empty set is indeed contained in any other set. The condition that every element of the empty set is contained in \(A\) is always satisfied, because the empty set doesn't have any elements. So there is no restriction.

Patrick Corn - 1 month ago

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I later thought so, thank you for your answer, though i wonder if it can be proven !

Andrew Hucek - 1 month ago

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