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109_sp2011_ho_ifsets_sol

# 109_sp2011_ho_ifsets_sol - 109 Spring 2011 Implication and...

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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 definition 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. First, we assume that a divides c . So by definition (as above), there is an integer m such that c = ma . Then 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 second implication is true. Since we have prove that both implications are true, their conjunction is also true. Exercise (I.5) . Which of the following conditions are necessary for the positive integer n to be divisible by 6? Which are suﬃcient? (i) 3 divides n (ii) 9 divides n (iii) 12 divides

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109_sp2011_ho_ifsets_sol - 109 Spring 2011 Implication and...

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