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Home » Let be the set of all rational numbers. Define an operation on by a . Show that (i) is a binary operation on (ii) is Commutative,
Let Q be the set of all rational numbers. Define an operation on Q-\{-1\} by a * b=a+b+a b. Show that (i) * is a binary operation on Q-\{-1\} (ii) { }^{*} is Commutative,
Binary Operations | CBSE | Class 12 | Exercise 3B | Maths | RS Aggarwal
Let Q be the set of all rational numbers. Define an operation on Q-\{-1\} by a * b=a+b+a b. Show that (i) * is a binary operation on Q-\{-1\} (ii) { }^{*} is Commutative,

(i)

* is an operation as a^{*} b=a+b+ ab where a, b \in Q-\{-1\} . Let a=1 and b=\frac{-3}{2} two rational numbers.

\mathrm{a}^{*} \mathrm{~b}=1+\frac{-3}{2}+1 \cdot \frac{-3}{2} \Rightarrow \frac{2-3}{2}-\frac{3}{2}=\frac{-1-3}{2} \Rightarrow \frac{-4}{2}=-2 \in \mathrm{Q}-\{-1\}

\mathrm{So}, * is a binary operation from \mathrm{Q}-\{-1\} \times \mathrm{Q}-\{-1\} \rightarrow \mathrm{Q}-\{-1\}

(ii)

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

\mathrm{b}^{*} \mathrm{a}=\frac{-3}{2}+1+\frac{-3}{2}, 1 \Rightarrow \frac{-3+2}{2}-\frac{3}{2}=\frac{-1-3}{2} \Rightarrow \frac{-4}{2}=-2 \in \mathrm{Q}-\{-1\}

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

i

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