Explanation Formula used: Definition of associativity: If ∗ is a binary operation on the non-empty set A , then ∗ is associative if x ∗ ( y ∗ z ) = ( x ∗ y ) ∗ z for all x , y , z in A . Proof: The multiplication in Q is defined by [ a , b ] ⋅ [ c , d ] = [ a c , b d ] . To show multiplication in Q is a binary operation, let [ a , b ] , [ c , d ] ∈ Q , such that b ≠ 0 , d ≠ 0 . Since D is an integral domain, b ≠ 0 and d ≠ 0 imply b d ≠ 0 , So [ a , b ] ⋅ [ c , d ] = [ a c , b d ] is an element of Q . To show that multiplication of two elements is unique (or well defined), suppose that [ a , b ] = [ x , y ] and [ c , d ] = [ z , w ] in Q . Claim: [ a , b ] ⋅ [ c , d ] = [ x , y ] ⋅ [ z , w ] . By using multiplication in Q , [ a , b ] ⋅ [ c , d ] = [ a c , b d ] and [ x , y ] ⋅ [ z , w ] = [ x z , y w ] Now [ a , b ] = [ x , y ] ⇒ a y = b x ……… (1) [ c , d ] = [ z , w ] ⇒ c w = d z …….. (2) To prove elements are equal, it is sufficient to show that ( a c ) ( y w ) = ( b d ) ( x z ) By multiplying the corresponding sides of equation (1) and (2), ( a y ) ( c w ) = ( b x ) ( d z ) By rearranging the terms, ( a c ) ( y w ) = ( b d ) ( x z ) Therefore, [ a c , b d ] = [ x z , y w ] ⇒ [ a , b ] ⋅ [ c , d ] = [ x , y ] ⋅ [ z , w ]

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ISBN: 9781285463230

Author: Gilbert, Linda, Jimmie

Publisher: Cengage Learning,

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Chapter 5.3, Problem 5E

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To prove: Multiplication is associative in Q.

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To prove: Multiplication is associative in Q .

Solution Summary: The author proves that multiplication is associative in Q — ast is a binary operation on the non-empty set A.

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