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Question 6. Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(v) Which elements of N are invertible for the operation ∗?

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Q. 13 Consider a binary operation ∗ on N defined as a * b = a^3 + b^3 . Choose the
correct answer.

(A) Is ∗ both associative and commutative?
(B) Is ∗ commutative but not associative?
(C) Is ∗ associative but not commutative?
(D) Is ∗ neither commutative nor associative?

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Q. 12State whether the following statements are true or false. Justify. 

(ii) If ∗ is a commutative binary operation on N, then a *(b*c) =( c*b)*a

(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.

Q.12 State whether the following statements are true or false. Justify.

(i) For an arbitrary binary operation ∗ on a set N,a*a = a,\; \forall a\in N

(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.    

Q. 11 Let A = N \times N and ∗ be the binary operation on A defined by
      (a, b) * (c, d) = (a + c, b + d)

Show that ∗ is commutative and associative. Find the identity element for ∗ on
A, if any.

 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.              

Q. 10 Find which of the operations given above has identity.

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.

Q. 9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(vi) a* b = ab^2

Find which of the binary operations are commutative and which are associative.

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.

Q. 9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(v) 

a * b = \frac{ab}{4}

Find which of the binary operations are commutative and which are associative.

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.

Q.9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iv) a * b = (a-b)^2

Find which of the binary operations are commutative and which are associative.

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.

Q.9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(iii) a * b = a + ab

Find which of the binary operations are commutative and which are associative.

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.

Q. 9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(ii) a*b = a^2 + b^2

Find which of the binary operations are commutative and which are associative.

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.                           

Q.9 Let ∗ be a binary operation on the set Q of rational numbers as follows:

(i)  a * b = a - b

Find which of the binary operations are commutative and which are associative.

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.                           

Q. 8 Let ∗ be the binary operation on N defined by a ∗ b = H.C.F. of a and b.
Is ∗ commutative? Is ∗ associative? Does there exist identity for this binary
operation on N?

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.    

Q. 7 Is ∗ defined on the set \{1, 2, 3, 4, 5\} by a * b = L.C.M. \;of \;a\; and \;b a binary
operation? Justify your answer.

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.          

Q.6 Let ∗ be the binary operation on N given by a* b = L.C.M.\; of \;a\; and \;b. Find 

(iv) the identity of ∗ in N

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

Q.6 Let ∗ be the binary operation on N given by a ∗ b = L.C.M. of a and b. Find

(iii) Is ∗ associative?

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

Q.6 Let ∗ be the binary operation on N given by a * b = L.C.M. \;of \;a\; and \;b. Find 

(ii) Is ∗ commutative?

(ii)     for all    Hence, it is commutative.  

Q.6 Let ∗ be the binary operation on N given by a * b = L.C.M. \;of\; a\; and \;b. Find

(i)  5 ∗ 7, 20 ∗ 16

a*b=LCM of a and b (i)  5 ∗ 7, 20 ∗ 16

Q.5 Let ∗′ be the binary operation on the set \{1, 2, 3, 4, 5\} defined by
a *' b = H.C.F. \;of\;a\;and\;b . Is the operation ∗′ same as the operation ∗ defined
in Exercise 4 above? Justify your answer.

   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.

Q.4 Consider a binary operation ∗ on the set {\{1, 2, 3, 4, 5\} given by the following
multiplication table (Table 1.2).

(iii) Compute (2 ∗ 3) ∗ (4 ∗ 5).

(Hint: use the following table)

(iii) (2 ∗ 3) ∗ (4 ∗ 5). from the above table  
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