Answer:
(i)
Given a * b = a ? b
Here
a * b = a ? b and b * a = b ? a
But
a ? b
b ?
is not
commutative.
Also,
=
a ? b ? c
and
is not
associative.
(ii) Given
Here
is
commutative.
Also
(a * b) c = (ab + 1) * c
=
(ab + 1) c + 1 = abc + c + 1
and
a * (b * c) = a * (bc + 1)
=
a(bc + 1) + 1 = abc + a + 1
is not
associative.
(iii)
Given
Here
is
commutative.
Also,
and
is
associative.
(iv) Given
Here a * b = 2ab
= 2ba
= b * a
is
commutative.
Also, (a * b) * c
(2ab) * c =
and
Clearly
is not
associative.
(v) Given a * b = ab
Here a * b = ab
and b * a = ba
is not
commutative.
Also (a * b) * c=
(ab) * (ab)c = abc
and a * (b * c) =
a * bc =
is not
associative.
(vi) Given
Here
Clearly a * b is not
necessarily equal to b * a. For example,, let a = 2 and b = 3
and
Hence a * b
is not
commutative.
Also
and
Clearly, a * (b
* c) is not necessarily equal to (a*b)*c
For example.
Let a = 2, b = 3
and c = 4
and
Hence * is not
associative.
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