Let Q be the set of all positive rational numbers.

Question:

Let Q be the set of all positive rational numbers.

(i) Show that the operation $*$ on $\mathrm{Q}^{+}$defined by $\mathrm{a} * \mathrm{~b}=\frac{1}{2}(\mathrm{a}+\mathrm{b})$ is a binary operation.

(ii) Show that $*$ is commutative.

(iii) Show that $*$ is not associative.

 

Solution:

(i) Let $a=1, b=2 \in Q+$

$a * b=\frac{1}{2}(1+2)=1.5 \in Q+$

* 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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