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Wk2-Abbott - Solutions to Week 2 Homework problems from...

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Solutions to Week 2 Homework problems from Abbott Problems (section 1.2) 1.3.1, 1.3.4, 1.3.6, 1.3.9, 1.4.2, 1.4.5, 1.4.11, 1.5.8 1.3.1 Note: This exercise is good practice for abstract algebra too! (a) Show that, given any element z Z 5 , there exists an element y such that z + y = 0. The element y is called the additive inverse of z . (b) Show that, given any z = 0 in Z 5 , there exists an element x such that zx = 1. The element x is called the multiplicative inverse of z . (c) Investigate the set Z 4 = { 0 , 1 , 2 , 3 } (where addition and multiplication are defined modulo 4) for the existence of additive and multiplicative inverses. Make a conjectuve about the values of n for which additive inverses exist in Z n , and then form another conjecture about the existence of multiplicative inverses. (a) 0 + 0 = 0; 1 + 4 = 0; 2 + 3 = 0 (addition is mod 5). Hence we have found additive inverses for each element of Z 5 . (b) 1 × 1 = 1; 2 × 3 = 1; 4 × 4 = 1 (multiplication mod 5). Hence we have multipliative inverses for each non-zero element. (c) 0 + 0 = 0; 1 + 3 = 0; 2 + 2 = 0 (all mod 4) hence additive inverses exist. But, 1 × 1 = 1; 3 × 3 = 1 (mod 4) but 2 does not have a multiplicative inverse. In fact, n needs to be a prime for Z n to be a field. 1.3.4 Assume that A and B are nonempty, bounded above, and satisfy B A . Show that sup B sup A . Proof: By definition of sup, we have sup A a for all a A , and sup A is the smallest number with this property. Suppose sup B > sup A . Then sup A cannot be an upper bound of B
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