109_sp2011_ho_ifsets_sol

109_sp2011_ho_ifsets_sol - 109 Spring 2011 - Implication...

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109 Spring 2011 - Implication and Subsets - Solutions Exercise (I.4) . Prove the following statements concerning positive integers a, b, and c . (i) ( a divides b )and( a divides c )= a divides ( b + c ). By deFnition of division, the assumption says that there are integers m, n such that b = ma and c = na. Then b + c = ma + na = a ( m + n ). Since m + n is an integer, this means that a divides b + c , as required. (ii) ( a divides b )or( a divides c )= a divides bc . We recall that a proposition of the form ( P 1 or P 2 )= Q is equivalent to the proposition ( P 1 = Q )and( P 2 = Q ). Thus, we prove the two implications. ±irst, we assume that a divides c . So by deFnition (as above), there is an integer m such that c = ma .Th en bc = b ( ma )= a ( bm )and a divides bc as required. To prove the second implication, we assume that b divides c . Analogously to before, there is an integer n such that b = na and hence bc = a ( cn ). Thus, a divides bc and the
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This note was uploaded on 08/02/2011 for the course MATH 109 taught by Professor Knutson during the Spring '06 term at UCSD.

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109_sp2011_ho_ifsets_sol - 109 Spring 2011 - Implication...

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