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 = 2^ab.
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:
(1*2)*3=2^(1*2)*3 = 4*3=2^(4*3)=2¹²
1*(2*3)=1*2x3=1*2 6=1*64=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: 
1*2=1² =1 and 2*1=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.