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If * is a binary operation in N defined ...

If * is a binary operation in N defined as a*b`=a^(3)+b^(3)` , then which of the following is true :
(i) * is associative as well as commutative.
(ii) * is commutative but not associative
(iii) * is associative but not commutative
(iv) * is neither associative not commutative.

Text Solution

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In N, `a * b = a^(3) + b^(3)`
Let `a, b in N`
`a* b = a^(3) +b^(3)`
`= b^(3) + a^(3) = b * a`
`therefore ` Operation * is commutative.
Again, let a, b,c `in`N.
`therefore a* (b *c) = a(b^(3) + c^(3))`
`" "=a^(3) + (b^(3) + c^(3))^(3)`
`and (a*b) *c = (a^(3) + b^(3)) *c `
` " "= (a^(3) + b^(3))^(3) + c^(3)`
`therefore" " a* (b*c) ne (a*b)*c`
`rArr` Operation * is not associative.
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