Let Q be the set of all positive rational numbers. (i) Show that the

1.	Let Q be the set of all positive rational numbers. (i) Show that the operation * on Q + defined \(ab=\frac{1}{2}(a+b)\) by is a binary operation. (ii) Show that is commutative. (iii) Show that * is not associative.
Answer» (i)Let a = 1, b = 2 ∈ Q ⁺ ab = \(\frac{1}{2}(1+2)\)= 1.5 ∈ Q⁺ is closed and is thus a binary operation on Q⁺ (ii) ab = \(\frac{1}{2}(1+2)\) = 1.5 And ba = \(\frac{1}{2}(2+1)\) = 1.5 Hence * is commutative. (iii)let c = 3. (ab)c = 1.5c = \(\frac{1}{2}(1.5+3)=2.75\) a(bc) = a\(\frac{1}{2}(2+3)\) = 12.5 = \(\frac{1}{2}(1+2.5)\) = 1.75 hence is not associative.**

Let Q be the set of all positive rational numbers. (i) Show that the operation * on Q + defined \(ab=\frac{1}{2}(a+b)\) by is a binary operation. (ii) Show that is commutative. (iii) Show that * is not associative.

Answer»

(i)Let a = 1, b = 2 ∈ Q ⁺

a*b = \(\frac{1}{2}(1+2)\)= 1.5 ∈ Q⁺

* is closed and is thus a binary operation on Q⁺

(ii) a*b = \(\frac{1}{2}(1+2)\) = 1.5

And b*a = \(\frac{1}{2}(2+1)\) = 1.5

Hence * is commutative.

(iii)let c = 3.

(a*b)*c = 1.5*c = \(\frac{1}{2}(1.5+3)=2.75\)

a*(b*c) = a*\(\frac{1}{2}(2+3)\) = 1*2.5 = \(\frac{1}{2}(1+2.5)\) = 1.75

hence * is not associative.

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Let Q be the set of all positive rational numbers. (i) Show that the operation * on Q + defined \(a*b=\frac{1}{2}(a+b)\) by is a binary operation. (ii) Show that * is commutative. (iii) Show that * is not associative.

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Let Q be the set of all positive rational numbers. (i) Show that the operation * on Q + defined \(ab=\frac{1}{2}(a+b)\) by is a binary operation. (ii) Show that is commutative. (iii) Show that * is not associative.