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Let * be the binary operation on the rational number given by a*b=a+b+2ab. which of the following are true?
i. * is commutative
ii. there is a rational number that is an identity with * operation
iii. every rational numebr has an inverse with * operation

I know that i is true and iii is false but why is ii false?

Ans: (i) is only true

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+1 vote
Both 1. and 2. should be true.

For identity e, a*e=e*a=e

a*e = a+e+2ae = a => e=0

e*a = e+a+2ea = a => e=0

So e=0 will be identity.
answered by Active (1.9k points)
they said that when a=-1/2 then no identity element exists. is it true?
@vijay

for e to be identity,

a*e = a

a+e+2ae = a

e+2ae = 0

e(1+2a) = 0

So either e = 0 or a = -1/2 . So when a = -1/2 then e can be anything hence no identity elem exist.
then e can be anything hence no identity elem exist.>what does it means ?

here 0 is identity element . But -1/2 also for some value .
Identity element should b unique so due to thiz II is false .

@vijay how u know iii is false .