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. Let * be a binary operation on $Q–{-1}$ defined by $a*b=a+b+ab$ for all $a,b\in Q-\left\{ -1 \right\}$. Then, (i) Show that * is both commutative and associative on $Q–{-1}$ (ii) Find the identity element in $Q–{-1}$
. Let * be a binary operation on $Q–{-1}$ defined by $a*b=a+b+ab$ for all $a,b\in Q-\left\{ -1 \right\}$. Then, (i) Show that * is both commutative and associative on $Q–{-1}$ (ii) Find the identity element in $Q–{-1}$
Binary Operations | CBSE | Class 12 | Exercise 3.4 | Maths | RD Sharma
. Let * be a binary operation on $Q–{-1}$ defined by $a*b=a+b+ab$ for all $a,b\in Q-\left\{ -1 \right\}$. Then, (i) Show that * is both commutative and associative on $Q–{-1}$ (ii) Find the identity element in $Q–{-1}$

(i) Firstly we all have to check commutativity of *

Assume $a,b\in Q-\left\{ -1 \right\}$

So, $a*b=a+b+ab$

$=b+a+ba$

$=b*a$

Hence,

$a*b=b*a$, $\forall a,b\in Q-\left\{ -1 \right\}$

Then we have to prove associativity of *

Assume $a,b,c\in Q-\left\{ -1 \right\}$, Now,

$a*(b*c)=a*(b+c+bc)$

$=a+(b+c+bc)+a(b+c+bc)$

$=a+b+c+bc+ab+ac+abc$

$(a*b)*c=(a+b+ab)*c$

$=a+b+ab+c+(a+b+ab)c$

$=a+b+ab+c+ac+bc+abc$

Hence,

$a*\left( b*c \right)=\left( a*b \right)*c,\forall a,b,c\in Q-\left\{ -1 \right\}$

So, * is associative on $Q–{-1}$.

(ii) Assume ‘e’ be the identity element in ${{I}^{+}}$ with respect to * such that

$a*e=a=e*a,\forall a\in Q-\left\{ -1 \right\}$

$a*e=a$and $e*a=a,\forall a\in Q-\left\{ -1 \right\}$

$a+e+ae=a$and $e+a+ea=a,\forall a\in Q-\left\{ -1 \right\}$

$e+ae=0$and $e+ea=0,\forall a\in Q-\left\{ -1 \right\}$

$e\left( 1+a \right)=0$and $e\left( 1+a \right)=0,\forall a\in Q-\left\{ -1 \right\}$

$e=0,\forall a\in Q-\left\{ -1 \right\}$ [because a not equal to $-1$]

So, $0$ is the identity element in $Q–{-1}$ with respect to *.

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