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

(ii) Is ∗ commutative?

(Hint: use the following table)

(ii)  For every  , we have . Hence it is commutative.

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

(i) Compute (2 * 3) * 4 and 2 * (3 * 4)

(Hint: use the following table)

(i)  

 Q.3 Consider the binary operation \wedge on the set \{1, 2, 3, 4, 5\} defined by
a \wedge b = min \{a, b\}. Write the operation table of the operation \wedge .

  for   The operation table of the operation  is given by : 1 2 3 4 5 1 1 1 1 1 1 2 1 2 2 2 2 3 1 2 3 3 3 4 1 2 3 4 4 5 1 2 3 4 5  

Q.2 For each operation ∗ defined below, determine whether ∗ is binary, commutative
or associative.

(vi) On R - \{-1 \}, definea * b = \frac{a}{b +1}

(iv) On , define                     and                                          for                the operation is not commutative.                                                        where    operation * is not associative.   

Q.2 For each operation ∗ defined below, determine whether ∗ is binary, commutative
or associative.

(v) On Z^+ , define a * b = a^b

(v) On , define                    and                                          for                the operation is not commutative.                                                        where    operation * is not  associative.   

Q.2 For each operation ∗ defined below, determine whether ∗ is binary, commutative
or associative.

(iv) On Z^+ , define a * b = 2^{ab}

(iv) On , define             ab = ba for all                   2ab = 2ba  for all                        for                the operation is commutative.                                                        where    operation * is not  associative.   

Q.2 For each operation ∗ defined below, determine whether ∗ is binary, commutative
or associative.

(iii) On Q, define a * b = \frac{ab}{2}

(iii) On , define                     ab = ba for all                            for all                        for   operation * is commutative.                                                          operation * is  associative.   

Q.2 For each operation ∗ defined below, determine whether ∗ is binary, commutative
or associative.

(ii) On Q, define a * b = ab + 1

(ii) On , define                    ab = ba for all                   ab+1 = ba + 1 for all                        for                                                         where    operation * is not associative.     

Q2. For each operation ∗ defined below, determine whether ∗ is binary, commutative
or associative.

(i)On Z, define a * b = a-b

a*b=a-b b*a=b-a so * is not commutative (a*b)*c=(a-b)-c a*(b*c)=a-(b-c)=a-b+c (a*b)*c not equal to a*(b*c), so * is not associative

Q.1 Determine whether or not each of the definition of ∗ given below gives a binary
operation. In the event that ∗ is not a binary operation, give justification for this

(v) On  Z^+ , define ∗ by a * b = a

(v) On   , define ∗ by  * carries each pair   to a unique element  in  . Therefore,* is a binary operation.

Q.1 Determine whether or not each of the definition of ∗ given below gives a binary
operation. In the event that ∗ is not a binary operation, give justification for this.

(iv) On Z^+, define ∗ by a * b = | a - b |

(iv) On , define ∗ by   We can observe that for ,there is a unique element   in . This means * carries each pair   to a unique element  in . Therefore,* is a binary operation.  

Q.1 Determine whether or not each of the definition of ∗ given below gives a binary
operation. In the event that ∗ is not a binary operation, give justification for this.

(iii) On R, define ∗ by a * b = ab^2

(iii) On R, define ∗ by a * b = ab^2

 We can observe that for a,b \in R,there is a unique element ab^{2}  in R.

This means * carries each pair (a,b)  to a unique element a * b = ab^{2} in R.

Therefore,* is a binary operation.

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Q. 1 Determine whether or not each of the definition of ∗ given below gives a binary
operation. In the event that ∗ is not a binary operation, give justification for this.

(ii) On Z^+ , define ∗ by a * b = ab

(ii) On , define ∗ by  We can observe that for ,there is a unique element ab in . This means * carries each pair   to a unique element  in . Therefore,* is a binary operation.  

Q.1 Determine whether or not each of the definition of ∗ given below gives a binary
operation. In the event that ∗ is not a binary operation, give justification for this.

(i) On Z^+ , define ∗ by a * b = a - b

(i) On , define ∗ by   It is not a binary operation as the image of  under * is  .                                                                                                                      
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