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Show that none of the operations given below has identity.(i) `a ** b = a-b ` (ii) `a ** b = a^2 + b^2 ` (iii) `a ** b = a +ab ` (iv) ` a ** b = (a-b)^2 ` (v) ` a ** b = (ab)/4` (vi) ` a**b = a b^2`

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(i)Given `a ** b = a-b `
`e` is the identity of `**` if
`a**e=e**a=a`
i.e,. `a-e=e-a=a`
...
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Let ** be the binary operation defined on Q . Find which of the following binary operations are commutative (i) a ** b=a-b, AA a,b in Q " " (ii) a ** b=a^(2)+b^(2), AA a,b in Q (iii) a ** b=a+ab, AA a,b in Q " " (iv) a ** b=(a-b)^(2), AA a,b in Q

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For each binary operation * defined below, determine whether * is commutative or associative. (i) On Z , define a ∗ b = a – b (ii) On Q , define a ∗ b = ab + 1 (iii) On Q , define a ∗ b = (ab)/(2) (iv) On Z^+ , define a ∗ b = 2^(ab) (v) On Z^+ , define a ∗ b = a^(b) (vi) On R – {– 1} , define a ∗ b = (a)/(b+1)

If a^(2) + b^(2) = 41 and ab = 4 , find : (ii) a+b