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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 $R - \{-1 \}$, define$a * b = \frac{a}{b +1}$

(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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