ABSTRACT MATH: How to show that an operation has no identity? Question shown as in textbook in the description (photo).

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It has to do with the concept of algebraic identities, the odd symbol that looks like a multiply is just a binary operation.

1 Answer
Mar 2, 2018

See explanation...

Explanation:

I will use #ox# for the binary operation.

Given:

#a ox b = a^2b+ab^2#

Suppose #e in RR# is an identity for #ox# and #a != 0# is any non-zero real number.

Then:

#a = a ox e = a^2e+ae^2 = a(a+e)e#

Dividing both ends by #a# and transposing, we find:

#(a+e)e = 1#

Hence:

#a = 1/e-e#

But #a# is arbitrary, so this will not hold generally.

Hence we can deduce that there is no such identity #e#.