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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions
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Let $\ast $ be the binary operation defined on Q.Find which of the following binary operations are commutative\begin{array}{1 1}(i)\;a \ast b=a-b\quad a,b\in Q & (ii)\;a \ast b=a^2+b^2\quad a,b\in Q\\(iii)\;a \ast b=a+ab\quad a,b\in Q & (iv)\;a \ast b=(a-b)^2\quad a,b\in Q\end{array}

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  • a binary operation * defined on Q in commutative if $ a*b=b*a$
Step1:
(i) $a*b=a-b\qquad a,b \in Q$
 
but $b*a =b-a \neq a-b \qquad a,b \in Q$
 
Hence * operation is not commutative
 
Step2:
(ii)$a*b =a^2+b^2 \qquad a,b \in Q$
 
$b *a=b^2+a^2=a^2+b^2$
 
[addition is commutaive in Q]
 
$a *b =b*a$
 
* operation is commutative
 
Step3:
(iii)$a*b =a+ab \qquad a,b \in Q$
 
But $b *a=b+ba$
 
$a*b \neq b*a$
 
* operation is not commutative
 
Step4:
(iv)$a*b=(a-b)^2\qquad a,b \in Q$
 
$b*a=(b-a)^2$
 
$=(-(a-b))^2$
 
$=(a-b)^2$
 
$a*b =b*a$
 
* operation is commutative
 
Solution:* operation defined by (ii) and (iv) are commutative

 

 

answered Mar 4, 2013 by meena.p
edited Mar 27, 2013 by meena.p
 

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