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Home » Let be the set of all positive rational numbers. (i) Show that the operation * on defined by is a binary operation. (ii) Show that * is commutative.
Let \mathrm{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.
Binary Operations | CBSE | Class 12 | Exercise 3B | Maths | RS Aggarwal
Let \mathrm{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.

(i) * is an operation as \mathrm{a}^{*} b=\frac{1}{2}(a+b) where a, b \in Q^{+}. Let a=1 and

b=2 two integers.

\mathrm{a}^{*} \mathrm{~b}=\frac{1}{2}(1+2) \Rightarrow \frac{3}{2} \in \mathrm{Q}^{+}

So, { }^{*} is a binary operation from \mathrm{Q}^{*} \times \mathrm{Q}^{*} \rightarrow \mathrm{Q}^{+}.

(ii) For commutative binary operation, \mathrm{a}^{*} \mathrm{~b}=\mathrm{b}^{*} \mathrm{a}.

\mathrm{b}^{*} \mathrm{a}=\frac{1}{2}(2+1) \Rightarrow \frac{3}{2} \in \mathrm{Q}^{+}

Since a^{*} b=b^{*} a, hence ^{*} is a commutative binary operation.

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