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Unformatted text preview: M114S, Solutions to HW #4 x4.11. The associativity of cardinal addition is shown on p. 43 and Exercise 4.27. Commutativity boils down to the assertion that κ ⊎ afii9838 = c afii9838 ⊎ κ . This commutativity property of disjoint unions holds for all sets, not just cardinals. It is proved by considering the map which interchanges blue and white in the first component of each element of a disjoint union. x4.12. We reduce the problems of cardinal arithmetic to familiar facts about disjoint unions, products, and functions. The associativity of cardinal multiplication stems from the associativity of products. In general A × ( B × C ) = c ( A × B ) × C via the map ( a, ( b, c )) mapsto→ (( a, b ) , c ). The commutativity of multiplication is due to the same property of products of sets, and A × B = c B × A via ( a, b ) mapsto→ ( b, a ). The distributive law for sets reads A × ( B ⊎ C ) = c ( A × B ) ⊎ ( A × C ). The map this time is ( a, ( i, x )) mapsto→ ( i, ( a, x )). Note that here i is either one of the tags for the disjoint union, and we use it on both sides of the equation. x4.13. We have already shown that P ( A ) = c ( A → { , 1 } ), for any set A ; in particular, for any cardinal number κ (which is a set), P ( κ ) = c ( κ → { , 1 } ), and hence P ( κ )  = c  ( κ → { , 1 } )  = c 2 κ , by the definition of the exponentiation operation on cardinals. x4.15. The correspondence between afii9839 → ( κ × afii9838 ) and ( afii9839 → κ ) × ( afii9839 → afii9838 ) is f mapsto→ ( f 1 , f 2 ), where f 1 ( x ) = First( f ( x )) and f 2 ( x ) = Second( f ( x ))....
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This homework help was uploaded on 04/16/2008 for the course MATH 114S taught by Professor Moschovakis during the Winter '08 term at UCLA.
 Winter '08
 MOSCHOVAKIS
 Set Theory, Addition, Sets

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