Question

For each binary operation * defined below, determine whether * is commutative or associative.
(i) On Z , define a * b=a−b
(ii) On Q , define a * b=ab+1
(iii) On Q , define a * b= $\frac {ab}{2}$.
(iv) On Z +, define a * b=2ab
(v) On Z +, define a * b=ab
(vi) On R −{−1},define a * b= $\frac {a}{b+1}$

Answer 1

(i) On Z , * is defined by a * b = a − b.
It can be observed that 1 * 2 = 1 − 2 = 1 and 2 * 1 = 2 − 1 = 1.
∴1 * 2 ≠ 2 * 1; where 1, 2 ∈ Z
Hence, the operation * is not commutative.
Also we have:
(1 * 2) * 3 = (1 − 2) * 3 = −1 * 3 = −1 − 3 = −4
1 * (2 * 3) = 1 * (2 − 3) = 1 * −1 = 1 − (−1) = 2
∴(1 * 2) * 3 ≠ 1 * (2 * 3) ;
where 1, 2, 3 ∈ Z

Hence, the operation * is not associative.

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(ii) On Q , * is defined by a * b = ab + 1.
It is known that: ab = ba ∀ a, b ∈ Q ⇒ ab + 1 = ba + 1∀ a, b ∈ Q ⇒ a * b = a * b ∀a, b ∈ Q
Therefore, the operation * is commutative.
It can be observed that:
(1 * 2) * 3 = (1 × 2 + 1) * 3 = 3 * 3 = 3 × 3 + 1 = 10
1 * (2 * 3) = 1 * (2 × 3 + 1) = 1 * 7 = 1 × 7 + 1 = 8
∴(1 * 2) * 3 ≠ 1 * (2 * 3) ;
where 1, 2, 3 ∈ Q

Therefore, the operation * is not associative.

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(iii) On Q , * is defined by a * b = $\frac{ab}{2}.$
It is known that:
ab = ba ∀ a, b ∈ Q $\frac {ab}{2}=\frac{ba}{2}$ ∀ a, b ∈ Q
⇒ a * b = b * a ∀ a, b ∈ Q
Therefore, the operation * is commutative.
For all a, b, c ∈ Q,
we have:
$(a*b)*c=(\frac{ab}{2})*c=\frac{(\frac{ab}{2})}{\frac{c}{2}}=\frac{abc}{4}$.
$a*(b*c)=a*(\frac{bc}{2})=\frac{a(\frac{bc}{2})}{2}=\frac{abc}{4}$.
Therefore (a * b) * c =a * (b * c)
Therefore, the operation * is associative.

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(iv) On Z +, * is defined by a * b = 2ab.
It is known that:
ab = ba ∀ a, b ∈ Z +
⇒ 2ab = 2ba ∀ a, b ∈ Z +
⇒ a * b = b * a ∀ a, b ∈ Z**+**
Therefore, the operation * is commutative.
It can be observed that:
(12)3=2(12)3 = 43=2(43)=212
1*(23)=12x3=12 6=164=2 64.
∴(1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈ Z+
Therefore, the operation * is not associative.

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(v) On Z ++ * is defined by a * b = ab.
It can be observed that:
12=12 =1 and 21=21=2
∴ 1 * 2 ≠ 2 * 1 ; where 1, 2 ∈ Z +
Therefore, the operation * is not commutative.
It can also be observed that:
$(2*3)*4=2^3*4=8*4=8^4=2^{12}$
$2(3*4)=2*3^4=2*81=2^{81}$
∴(2 * 3) * 4 ≠ 2 * (3 * 4) ; where 2, 3, 4 ∈ Z +
Therefore, the operation * is not associative.

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(vi) On R , * − {−1} is defined by $a*b=\frac {a}{b+1}.$
It can be observed that $1*2=\frac{1}{2+1}=\frac{1}{3} and 2*1=\frac{2}{1+1}=\frac{2}{2}=1$.
∴1 * 2 ≠ 2 * 1 ; where 1, 2 ∈ R − {−1}
Therefore, the operation * is not commutative.
It can also be observed that:
$(1*2)*3=\frac{\frac{1}{3}}{3+1}=\frac{1}{12}.$
$1*(2*3)=1*\frac{2}{3+1}=1*\frac{2}{4}=\frac{1}{\frac{1}{2+1}}=\frac{1}{3}=\frac{2}{3}$
∴ (1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈ R − {−1}
Therefore, the operation * is not associative.