A binary operation is defined on the set x = R – – 1 by x y = x +… - Vidyadip

Question

A binary operation $\ast$ is defined on the set x = R – { – 1 } by
$ \text{x}\ast \text{y} = \text{x + y + xy,}\forall \text{x, y}\in \text{X}.$
Check whether $\ast$ is commutative and associative. Find its identity element and also find the inverse of each element of X.

✓

Answer

$\text{(i) commutative : let x, y}\in \text{R} -\left\{-1\right\} \text{then}$
$\text{x}\ast\text{y = x + y + xy + y + x + yx = y} \ast\text{x}\therefore \ast \text{ is commutative}$
$\text{(ii)Associative : let x, y, z}\in \text{R} -\left\{-1\right\} \text{then}$
$\text{x}\ast\text{(y}\ast\text{z = x}\ast\text{(y + z + yz) = x + ( y + z + yz) + x (y + z + yz)}$
$=\text{x + y + z + xy + yz + zx + xyz}$
$\text{(x}\ast\text{y})\ast\text{z = (x + y + xy)}\ast\text{z = (x + y + xy) + z + (x + y + xy). z}$
$=\text{x + y + z + xy + yz + zx + xyz}$
$\text{x}\ast(\text{y}\ast\text{z}) = \text{(x}\ast\text{y})\ast \text{ z}\therefore\ast \text{ is Associative}$
$\text{(iii)Identity Element : let e}\in \text{R}-\left\{-1\right\}\text{such that a}\ast\text{e = e}\ast \text{a = a}\forall \text{a}\in \text{R}\left\{-1\right\}$
$\therefore\text{a + e + ae = a}\Rightarrow\text{e = 0}$
$\text{(iv) Inverse : let a}\ast\text{b} = \text{b}\ast\text{a = e = 0: a, b}\in \text{R}- \left\{-1\right\}$
$\Rightarrow \text{a + b + ab = 0}\therefore\text{b}= \frac{\text{-a}}{\text{1 + a}} \text{or} \text{ a}^{-1} = \frac{\text{-a}}{\text{1 + a}} $

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