{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

HW3_199_sol

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online