Answer:
(i)
Given a * b = a ? b
Since
a ? b
is not
commutative.
Also
(a * b) * c = (a ? b) * c = a ? b ? c
and
a * (b * c) = a *(b ? c) = a ? (b ? c)
=
a ? b + c
As
a ? b ? c
a ? b + c
is not
associative.
(ii) Given,
a * b = a2 = b2,
a
* b = a2 + b2 = b2 + a2 = b* a
commutative.
Also
(a * b) * c = (a2 + b2) * c
=
(a2 + b2)2 + c2
and
a * (b * c) = a * (b2 + c2)
=
a2 + (b2 + c2)2
is
associative.
(iii) Given
a * b = a +
Here,
a * b = a + ab
And
b * a = b + ba.
But
a + ab
b + ba
But
a * b
b * a
is not
commutative.
Also
(a * b) * c = (a + ab) * c
=
a + ab + (a + ab) c
=
a + ab + ac + abc
and
a * (b * c) = a * (b + bc)
=
a + a (b + bc)
=
a + ab + abc
But
a + ab + ac + abc
a +
ab + abc
is not
associative.
(iv) Given,
a * b = (a ? b)2
Here a * b = (a ?
b)2 = (b ? a)2
= b * a
is
commutative.
Also, (a * b) * c =
(a ? b)2 * c
= ((a ? b)2
? c)2
and a * (b * c) =
a * (b ? c)2
= (b ? (b ? c)2)2
But ((a ? b)2
? c)2
is not
associative.
(b) Given a * b =
Here,
is
commutative.
Also,
and
is
associative.
(vi) Given a * b = ab2
= ab2
Now, a * b = ab2
and b * a = b2a
is not
commutative.
Also, (a * b) * c
= (ab2) * c = ab2 c2
and
a * (b * c) = a* (bc2)
= a(bc2)2
= ab2 c4
Clearly (a * b) *
c
is not
associative.
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