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HW3_199_sol - HOMEWORK 3 DUE Fri Oct 14 NAME DIRECTIONS •...

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Unformatted text preview: HOMEWORK 3 DUE: Fri., Oct. 14 NAME: DIRECTIONS: • Turn in your homework as SINGLE-SIDED typed or handwritten pages. • STAPLE your homework together. Do not use paper clips, folds, etc. • STAPLE this page to the front of your homework. • Be sure to write your name on your homework. • Show all work, clearly and in order . You will lose point 0.5 points for each instruction not followed. Questions Points Score 1 1 2 1 3 1 4 1 5 2 6 1 7 2 8 1 Total 10 1 HOMEWORK 3 DUE: Fri., Oct. 14 NAME: Problem 1: (1 point) (a) (0.5 point) Show that { ( − a, b ) } is an additive inverse for { ( a, b ) } . Proof: Consider { ( a, b ) } + { ( − a, b ) } = { ( ab + b ( − a ) , b 2 ) } = { (0 , b 2 ) } = { (0 , 1) } , which is the additive identity in Q . Q.E.D. (b) (0.5 point) Prove the distributive law for Q . Proof: Let { ( a, b ) } , { ( c, d ) } , and { ( e, g ) } ∈ Q . We want to show { ( a, b ) } · [ { ( a, b ) } + { ( e, g ) } ] = { ( a, b ) } · { ( c, d ) } + { ( a, b ) } · { ( e, f ) } . A little algebra shows that L.H.S. = { ( a, b ) } · { ( cf + de, df ) } = { ( acf + ade, bdf ) } , R.H.S = { ( ac, bd ) } + { ( ae, bf ) } = { ( acbf + bdae, b 2 df ) } = { ( b, b ) } · { ( acf + dae, bdf ) } , but { ( b, b ) } = { (1 , 1) } which is the multiplicative identity in Q so Q.E.D. Problem 2: (1 points) Let R be a ring and R a nonempty subset of R . Show that R is a subring iff, for any a , b ∈ R , we have a − b , ab ∈ R ....
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HW3_199_sol - HOMEWORK 3 DUE Fri Oct 14 NAME DIRECTIONS •...

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