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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS5 Marks
Question
$\text{Let A = Q} \times \text{Q}$ and let $*$ be a binary operation on A defined by$\text{(a, b)} $*$ \text{(c, d) = (ac, b + ad)} \text{ for (a, b), (c, d)} \in \text{A}.$ Determine, whether $*$ is commutative and associative. Then, with respect to $*$ on A.
  1. Find the identity element in A.
  2. Find the invertible elements of A.
✓

Answer

$\text{(a, b)}{*} \text{(c, d) = (ac, b + ad); (a, b), (c, d)} \in \text{A}$
$\text{(c, d)}{*} \text{(a, b) = (ca, d + bc)}$
Since $\text{b + ad} \neq \text{d + bc} \Rightarrow \text{ }{*}$is NOT comutative
for associativity, we have,
$\text{(a, b)}{*} \text{[(c, d)}{*} \text{(e, f)] = (a, b)}{*} \text{(ce, d + cf) = (ace, b + ad + acf)}$
$\Rightarrow \text{ }{*} \text{ is associative}$
Let (e, f) be the identity element in A
$\text{Then (a, b)}{*} \text{(e, f) = (a, b) = (e, f)}{*} \text{(a, b)}$
$\Rightarrow\text{(ae, b + af) = (a, b) = (ae, f + be)}$
$\Rightarrow \text{e = 1, f = 0} \Rightarrow \text{(1, 0) is the identity element}$
Let (c, d) be the inverse element for (a, b)
$\Rightarrow \text{(a, b)}{*} \text{(c, d) = (1, 0) = (c, d)}{*} \text{(a, b)}$
$\Rightarrow \text{(ac, b + ad) = (1, 0) = (ac, d + bc)}$
$\Rightarrow \text{ac} = 1 \Rightarrow \text{c} = \frac{1}{\text{a}} \text{ and b + ad} = 0 \Rightarrow \text{d} = - \frac{\text{b}}{\text{a}} \text{ and d + bc =0 } \Rightarrow \text{d = -bc = -b} \bigg(\frac{1}{\text{a}}\bigg)$
$\Rightarrow \bigg(\frac{1}{\text{a}}, - \frac{\text{b}}{\text{a}}\bigg), \text{a} \neq 0 \text{ is the inverse of (a, b)} \in \text{A}$

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