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matt grime
Jun26-05, 04:22 AM
Sci Advisor
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P: 9,396
As with a lot of proofs by contradiction the fist line "assume that something is not..." is not necessary.

f injective iff ker(f)={e} ( the => is easy since it is a special case of the the general property of being injective: f(e)=e so if anything else satisfies f(g)=e thetn g=e and ker(f)={e})

f(x)=f(y) implies f(xy^{-1}=e, which, by hypothesis means xy^{-1}=e as we required to show.

so we don't prove it by assuming the opposite and getting a contradiction since the proof is actually direct. the same can be done for cantor's diagonal argument.