On R, binary operation * is defined by a * b=a+b+a b then identity and inverse of * are ...... respectively.

Question

On R, binary operation * is defined by a * b=a+b+a b then identity and inverse of * are ...... respectively. (A) 0, (a/1-a) (B) 1, (a/1+a) (C) 0,-(a/1+a) (D) 1, (a/1-a)

Tardigrade Admin · Accepted Answer

If a* e=e* a=a then e is identity a+e+a e=a e(1+a)=0 e=0 (identity) Now if a* b=e then b is inverse of a a+b+a b=0 a=-b(1+a) inverse b=(-a/1+a)

Q. On $R$ , binary operation $$ is defined by $a b = a + b + ab$ then identity and inverse of $*$ are ...... respectively.

$0, \frac{a}{1 - a}$

$1, \frac{a}{1 + a}$

$0, - \frac{a}{1 + a}$

$1, \frac{a}{1 - a}$

If $a * e = e * a = a$ then $e$ is identity
$a + e + a e = a$
$e (1 + a) = 0$
$e = 0$ (identity)
Now if $a * b = e$ then $b$ is inverse of $a$
$a + b + ab = 0$
$a = - b (1 + a)$
inverse $b = \frac{- a}{1 + a}$

Q. On R, binary operation ∗ is defined by a∗b=a+b+ab then identity and inverse of ∗ are ...... respectively.

0,1−aa​

1,1+aa​

0,−1+aa​

1,1−aa​