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State whether the following statements a...

State whether the following statements are true or false. Justify. (i) For an arbitrary binary operation `*` on a set `N ," "a" "*" "a" "=" "a""""""AA""""""""a in N` . (ii) If `*` is a commutative binary operation on N, then `a" "*" "(b" "*" "c)"

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Let in N, the binary operation * is defined as a * b = a+ b
`therefore " " 2,3 in N`
`and " "2* 3 = 2+ 3 = 5 ne 2`
`therefore` Given statements is false
(ii) True
L.H.S = a *(b*c)
`" "=a *(c+b)" "(because * "is communtative")`
`" " = ( c*b) * a" "(because * " is commutative")`
`" " = R.H.S`
`therefore` Given statement is ture.
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