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I just realized some simple math too....

S

Semidevil

Diamond Member
Given any set S and T, if there exists an injection from S to T, and T to S, then S = T.


(everyone else is doing one tonight, I might as well do one too)
 
Consider S = {1, 2} and T = {shoe, apple, grape}. Define f: S -> T by f(1) = shoe and f(2) = apple. f is injective, but obviously S != T. Perhaps you meant that |S| = |T| (i.e., the sets have the same cardinality), but that's not true either (as per my example). You need to require that there is a bijection from S to T in order to have |S| = |T| (or equivalently: an injection from S to T, and an injection from T to S).
 
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