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).

  • $\begingroup$ 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. $\endgroup$ – saulspatz Oct 11 '18 at 16:30
  • $\begingroup$ This might help $\endgroup$ – Chinnapparaj R Oct 11 '18 at 16:31
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    $\begingroup$ 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. $\endgroup$ – Asaf Karagila Oct 11 '18 at 16:32
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    $\begingroup$ This might also help en.wikipedia.org/wiki/Vacuous_truth $\endgroup$ – Yanko Oct 11 '18 at 16:32
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    $\begingroup$ 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? $\endgroup$ – David K Oct 11 '18 at 17:33