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Consider a binary operation star on N defined as a star b equals a to the power 3 plus b to the power 3 . Choose the correct answer.(A) Is star both associative and commutative? (B) Is star commutative but not associative? (C) Is star associative but not

Q. 13 Consider a binary operation ∗ on N defined as a * b = a^3 + b^3 . Choose the
correct answer.

(A) Is ∗ both associative and commutative?
(B) Is ∗ commutative but not associative?
(C) Is ∗ associative but not commutative?
(D) Is ∗ neither commutative nor associative?

Answers (2)
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A binary operation ∗ on N defined as a * b = a^3 + b^3 .

For a,b \in N

       a * b = a^3 + b^3 = b^{3}+a^{3}=b*a

Thus , it is commutative.

(1*2)*3 = (1^{3}+2^{3})*3=9*3 =9^{3}+3^{3}=729+27=756

1*(2*3) = 1*(2^{3}+3^{3})=1*35 =1^{3}+35^{3}=1+42875=42876

\therefore \, \, \, \, \, (1*2)*3 \neq 1*(2*3)     where  1,2,3 \in N

Hence, it is not associative.

Hence, B is correct option.

 

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