1. Complete the proof in Lemma 5.4.3 to show thatmultiplication of equivalence c
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1. Complete the proof in Lemma 5.4.3 to show thatmultiplication of equivalence classes in Q(D) iswell-defined. Lemma 5.4.3 Let D be an integraldomain. Define binary operations + and . on Q(D)by [a,b] + [c+d] = [ad + bc, bd] and[a,b].[b,c] = [ac, bd], for [a,b], [c,d] in Q(D). Then + and . arewell defined operetions on Q(D). 1. Complete the proof in Lemma 5.4.3 to show thatmultiplication of equivalence classes in Q(D) iswell-defined. Lemma 5.4.3 Let D be an integraldomain. Define binary operations + and . on Q(D)by [a,b] + [c+d] = [ad + bc, bd] and[a,b].[b,c] = [ac, bd], for [a,b], [c,d] in Q(D). Then + and . arewell defined operetions on Q(D). [a,b] + [c+d] = [ad + bc, bd] and[a,b].[b,c] = [ac, bd], for [a,b], [c,d] in Q(D). Then + and . arewell defined operetions on Q(D).Explanation / Answer
To prove the multiplication of equivalent classes is welldefined we want to show if [a,b] ~[a',c'] and [c,d]~[c',d'] then [a,b].[c,d] ~[a',b'].[c',d'] (that is [ac, bd] =[a'c', b'd'] ) So we assume [a,b] ~[a',b'] and [c,d]~[c',d'] then ab' = ba' and cd' = c'd. Now we have ab'cd' = ba'c'd. acb'd' = bda'c'. Then by definition [ac, bd] = [a'c', b'd']. That is [a,b].[c,d] ~[a',b'].[c',d'].Related Questions
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