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For each operation ∗ defined below, determine whether ∗ is binary, commutative or associative. On Q, define a ∗ b = ab + 1

Q.2 For each operation ∗ defined below, determine whether ∗ is binary, commutative
or associative.

(ii) On Q, define a * b = ab + 1

Answers (1)
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(ii) On Q, define a * b = ab + 1

                   ab = ba for all a,b \in Q

                 ab+1 = ba + 1 for all a,b \in Q

\Rightarrow               a\ast b=b\ast a       for a,b \in Q

         (1*2)*3 = (1\times 2+1) * 3 = 3 * 3 = 3\times 3+1 = 10

            1*(2*3) = 1 * (2\times 3+1) = 1 * 7 = 1\times 7+1 = 8

              \therefore             (1\ast 2)\ast 3\neq 1\ast (2\ast 3) ;     where 1,2,3 \in Q

\therefore  operation * is not associative.   

 

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