A simple * question

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A simple * question

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whats the question
Develop a proof to show that \[a ° b =\frac{a+b}{ab}\]Has no identity (i.e. \(a ° b = a\)).
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Here is what I got
Is that ever true? Let's use some real numbers: 7 * 3 = (7 + 3) / 7*3 7*3 =? 10/21 21 =? 10/21
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
Why do we assume that a*b = a?
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
for all a belonging to the required set
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?
@Nishant_Garg, the \(ab\) should be at the bottom, not the top
oh wow my bad haha
Yeah, that changes a bit ;)
but it should be something like that if u just solve it
I don't know how to rearrange to get something similar, can you help @Nishant_Garg?
ill try
Thanks so much @Nishant_Garg :D
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
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?
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...
Yeah. Thanks so much @Nishant_Garg :D

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