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I am struggling to prove what seems like a trivial fact - subtracting a smaller set from a bigger set cannot produce the empty set - and it just seems like there must be a simple proof of this.

Is there a way to prove that if $|A| > |B|$, then $A - B \ne \emptyset$?


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up vote 7 down vote accepted

If $A\setminus B = \varnothing$, then $A \subseteq B$ so inclusion provides an injection from $A$ into $B$ implying $\lvert A \rvert \leq \lvert B \rvert$.

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That's beautiful. Thanks so much! – templatetypedef Jan 16 '13 at 4:44

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