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S seema garhwal
(ii) If ∗ is a commutative binary operation on N, then R.H.S                        (* is commutative)                          (  as * is commutative)              =    L.H.S             Hence, statement  (ii) is true.

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S seema garhwal
(i) For an arbitrary binary operation ∗ on a set N, An operation * on a set N as  Then , for  b=a=2 Hence, statement (i) is false.    

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S seema garhwal
 and ∗ be the binary operation on A defined by Let    Then,  We have        Thus it is commutative. Let    Then,  Thus, it is associative. Let   will be a element for operation *  if  for all . i.e.  This is not possible for any element in A . Hence, it does not have any identity.              

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S seema garhwal
An element   will be identity element for operation * if    for all           when    . Hence,   has identity as 4. However, there is no such element    which satisfies above condition for all rest five operations. Hence, only (v) operations have identity.

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S seema garhwal
On the set Q ,the operation * is defines as   .It is observed that: For            for   Hence, the * operation is not  commutative. It can be observed that     for all  The operation * is  not associative.

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S seema garhwal
On the set Q ,the operation * is defines as   .It is observed that: For            for   Hence, the * operation is commutative. It can be observed that     for all  The operation * is associative.

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S seema garhwal
On the set Q ,the operation * is defined as   .It is observed that: For            for   Hence, the * operation is commutative. It can be observed that     for all  The operation * is not associative.

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S seema garhwal
On the set Q ,the operation * is defines as   .It is observed that: For          for   Hence, the * operation is not  commutative. It can be observed that     for all  The operation * is not associative.

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S seema garhwal
On the set Q ,the operation * is defines as  .It is observed that: For          Hence, the * operation is commutative. It can be observed that      for all    The operation * is not associative.                           

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S seema garhwal
On the set Q ,the operation * is defines as  .It is observed that:              here  Hence, the * operation is not commutative. It can be observed that      for all    The operation * is not associative.                           

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S seema garhwal
a ∗ b = H.C.F. of a and b for all  H.C.F. of a and b = H.C.F of b and a for all  Hence, operation  ∗  is commutative. For   ,                      Hence,  ∗ is associative. An element  will be identity for operation *  if   for . Hence, the operation * does not have any identity in N.    

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S seema garhwal
A =  Operation table is as shown below: 1 2 3 4 5 1 1 2 3 4 5 2 2 2 6 4 10 3 3 6 3 12 15 4 4 4 12 4 20 5 5 10 15 20 5 From the table, we can observe that Hence, the operation is not a binary operation.          

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S seema garhwal
(iv) the identity of ∗ in N We know that                       for  Hence, 1 is the identity of ∗ in N.  

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S seema garhwal
a  b = L.C.M. of a and b (iii)                Hence, the operation is associative.  

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S seema garhwal
(ii)     for all    Hence, it is commutative.  

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S seema garhwal
   for   The operation table is as shown below: 1 2 3 4 5 1 1 1 1 1 1 2 1 2 1 2 1 3 1 1 3 1 1 4 1 2 1 4 1 5 1 1 1 1 5 The operation ∗′ same as the operation ∗ defined in Exercise 4 above.
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