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sols20 - 20. NOVEMBER 15TH: CARDINALITY II 47 20. November...

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20. NOVEMBER 15TH: CARDINALITY II 47 20. November 15th: Cardinality II 20.1 . Give an alternative proof of Lemma 20.1 by proving directly that the func- tion f : A B deFned in the proof given in § 20.1 is both injective and surjective (without using the inverse functions g S ). Answer: Recall that f is deFned by setting (for all a A ) f ( a )= f S ( a ), where S is the unique element of the partition P A containing a . To prove that f is injective, assume that f ( a )= f ( a ° )fo rtwoe l em en t s a , a ° of A .Inp a r t i cu l a r , f ( a )and f ( a ° )mustbe longtooneandthesamee lemento fthe partition P B ;ca l lth i se lemen t T .S i n c e α is a bijective map P A P B , T is the image of a unique element S of the partition P A .Byth eway f is deFned, a and a ° must both belong to S .S in c e f S is bijective, it is particular injective, and it follows that a = a ° .Th u s ,w eh a v esh ownth a t f ( a )= f ( a ° )imp l ies a = a ° ,andth isshows that f is injective. To prove that f
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sols20 - 20. NOVEMBER 15TH: CARDINALITY II 47 20. November...

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