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Binary Operations — MATHS STD 12 Science — Question

Rajasthan BoardEnglish MediumSTD 12 ScienceMATHSBinary Operations3 Marks
Question
Let * be a binary operation on Q - {-1} defined by a * b = a + b + ab for all a, b ∈ Q - {-1}. Then,
Show that every element of Q − {−1} is invertible. Also, find the inverse of an arbitrary element.
✓

Answer

We have,
a * b = a + b + ab for all a, b ∈ Q - {-1}
Let b be the inverse of a ∈ Q - {-1}
Then, a * b = b * a = e [e is the identity element]
⇒ a + b + ab = e
⇒ a + b + ab = 0
⇒ b(1 + a) = -a
$\Rightarrow\text{b}=\frac{-\text{a}}{1+\text{a}}$ $\begin{bmatrix}\because\ \frac{-\text{a}}{1+\text{a}}\neq-1\text{ because if }\frac{-\text{a}}{1+\text{a}}=-1\\\Rightarrow\text{a}=1+\text{a}\Rightarrow1=0\text{ Not possible}\end{bmatrix}$
$\text{b}=\frac{-\text{a}}{1+\text{a}}$ is the inversre of a with respect to *.

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