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.
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Expert's answer
2012-10-25T11:13:30-0400
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
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