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RELATIONS AND FUNCTIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsRELATIONS AND FUNCTIONS5 Marks
Question
Let $\text{A} = \text{R} \times \text{R}$ and let $*$ be a binary operation on A defined by $\text{(a, b)} {*} \text{(c, d)} = \text{(ad + bc, bd)}$ for all $\text{(a, b), (c, d)} \in \text{R} \times \text{R}.$
  1. Show that $*$ is commutative on A.
  2. Show that $*$ is associative on A.
  3. Find the identity element of $*$ in A.
✓

Answer

  1. $\text{(a, b)}{*} \text{(c, d) = (ad + bc, bd)}$
$\text{Now}, \text{(c, d)}{*} \text{(a, b) = (cb + da, db) = (ad + bc, bd) = (a, b)}{*} \text{(c, d)}$
$\Rightarrow \text{ }{*} \text{ is Communicative}$
  1. $\text{[(a, b) }{*} \text{(c, d)]}{*} \text{(e, f) = (ad + bc, bd)}{*} \text{(e, f) = (adf + bcf + bde, bdf)]}$
$\text{(a,b)}{*} \text{[(c, d)}{*} \text{(e, f)] = (a, b)} {*} \text{(cf + de, df) = (adf + bcf + bde, bdf)}$
$\Rightarrow \text{ } {*} \text{ is associative.}$
  1. Let $(e_1, e_2)$ be the identity element of A.
$\Rightarrow \text{(a, b}) {*} \text{(e}_{1}, {\text{e}_{2}) = \text{(a, b)} = \text{(e}_{1}, \text{e}_{2}) {*} \text{(a, b)}}$
$\Rightarrow \text{(ae}_{2} + \text{be}_{1} , \text{be}_{2}) = \text{(a, b)} = \text{(e}_{1}\text{b} + \text{e}_{2}\text{a}, \text{e}_{2} \text{b})$
$\Rightarrow \text{a}\text{e}_{2} + \text{b}\text{e}_{1} = \text{a and b}\text{e}_{2} = \text{b} \Rightarrow \text{e}_{1} = 0, \text{e}_{2} = 1$
$\Rightarrow$ (0, 1) is the identity on A.

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