Question

A simple * question

Answer 1

whats the question

Answer 2

Develop a proof to show that \[a ° b =\frac{a+b}{ab}\]Has no identity (i.e. \(a ° b = a\)).

Answer 3

Any clue?

Answer 4

Here is what I got

Answer 5

Is that ever true?
Let's use some real numbers:
7 * 3 = (7 + 3) / 7*3
7*3 =? 10/21
21 =? 10/21

Answer 6

Assume that \[a*b=a\]
then,
\[\frac{ab}{a+b}=a\]\[\frac{1}{a}+\frac{1}{b}=a\]\[\frac{1}{b}=a-\frac{1}{a}\]\[\frac{1}{b}=\frac{a^2-1}{a}\]\[b=\frac{a}{a^2-1}\]
Here b would be the identity element, for which all a*b=a
however this is not true as at a=1 b is not defined, therefore there is no identity element

Answer 7

Why do we assume that a*b = a?

Answer 8

If our assumption is correct, then we'd get an expression for b which is defined for all a, but since b is not defined at a=1, our assumption is incorrect and there's no identity element

Answer 9

for all a belonging to the required set

Answer 10

Ahh, that make's sense @Nishant_Garg. Thank you! :D.

Also, from my textbook, it says that there can only be one identity element. Since in this case, the identity element would change (the squared would affect it), is that another reason aswell?

Answer 11

@Nishant_Garg, the \(ab\) should be at the bottom, not the top

Answer 12

oh wow my bad haha

Answer 13

Yeah, that changes a bit ;)

Answer 14

but it should be something like that if u just solve it

Answer 15

I don't know how to rearrange to get something similar, can you help @Nishant_Garg?

Answer 16

ill try

Answer 17

Thanks so much @Nishant_Garg :D

Answer 18

ok there's only 1 mistake, i've written
\[\frac{ab}{a+b}\]
instead of
\[\frac{a+b}{ab}\]
but the rest of the calculation is ok
As for the identity element, yes there's only one identity element...so about that I'm also confused O.o

Answer 19

Maybe, what we are meant to do is to sub in some value for \(a\), and find the corresponding \(b\) vaule. Then, substitute anoter \(a\) value. If the \(b\) values don't match, then there is no identity element, @Nishant_Garg?

Answer 20

yep, and clearly they won't match! and it's not even defined for some numbers like +-1, so there r definitely many factors to it...

Answer 21

Yeah. Thanks so much @Nishant_Garg :D