# Answer to Question #17262 in Abstract Algebra for Tsit Lam

Question #17262

Ex. 4.1. In R, define a ◦ b = a + b − ab. Show that this binary operation is associative, and that (R, ◦) is a monoid with zero as the identity element.

Expert's answer

a ◦ b belongs to R

associative because

a ◦ (b ◦ c)=a ◦

(b+c-bc)=a+b+c-bc-a*(b+c-bc)=a+b+c-ab-ac-bc+abc

(a ◦ b) ◦ c = (a+b-ab) ◦

c=a+b-ab+c-(a+b-ab)c=a+b+c-ab-ac-bc+abc

zero is 0

a ◦

0=a+0-a*0=a

0 ◦ a=0+a-0*a=a

so R with operation ◦ is a monoid

associative because

a ◦ (b ◦ c)=a ◦

(b+c-bc)=a+b+c-bc-a*(b+c-bc)=a+b+c-ab-ac-bc+abc

(a ◦ b) ◦ c = (a+b-ab) ◦

c=a+b-ab+c-(a+b-ab)c=a+b+c-ab-ac-bc+abc

zero is 0

a ◦

0=a+0-a*0=a

0 ◦ a=0+a-0*a=a

so R with operation ◦ is a monoid

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