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I'm trying to understand why the empty set is a subset of every set and this is my reasoning (please correct me if I'm wrong):

By definition of a set S is a subset of a set A if all it’s elements are in A. ∅ has no elements if ∅ is not a subset of A then there is an element in ∅ that is not in A but ∅ as not elements. So ∅ is a subset of A by contradiction.

does it really follow that because the empty set has no elements it is a subset of every set? I mean imagine this: I have a box with nothing in it (the empty set). then I have a box with something in it (3 balls if you will). how is "nothingness" (if I may call the element of the empty set that) be in a box that has something in it already (the three balls).

Andrés E. Caicedo
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jessika
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  • The set of balls in the first box is a subset of the set of balls in the second set. The boxes are physical objects; sets are mathematical concepts with no physical existence. – saulspatz Oct 11 '18 at 16:30
  • This might help – Chinnapparaj R Oct 11 '18 at 16:31
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    The "sets are like boxes" analogy only takes you so far. How could a box contain you while another box contain you and me? Physical analogies are not meant to replace actual mathematical definitions. – Asaf Karagila Oct 11 '18 at 16:32
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    This might also help https://en.wikipedia.org/wiki/Vacuous_truth – Yanko Oct 11 '18 at 16:32
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    You could look at it this way: whenever you take things out of a set, you're left with a subset of what you started with. If you take everything out, what are you left with? – David K Oct 11 '18 at 17:33
  • For any set S, empty set is subset S, because every element of empty set is in S. Because you cannot find an element of empty set that is not in S. – William Elliot Oct 12 '18 at 07:14

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