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Prove that the empty set is a subset of every set.

I don't really know where to start other than the fact that I know a symbolic representation of the empty set and that it is included in every set.

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The definition of $A\subseteq B$ is that, for every element $x\in A$, it follows that $x\in B$ as well. Since the empty set has no elements, this is true vacuously.

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  • $\begingroup$ Similarly, all Unicorns have 2 horns. Proof: let A = the set of all Unicorns, and B be the set of animals with two horns. $\endgroup$ – Peter Webb Mar 3 '15 at 7:06

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