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Ahsome
 one year ago
A simple * question
Ahsome
 one year ago
A simple * question

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ahsome
 one year ago
Best ResponseYou've already chosen the best response.0Develop a proof to show that \[a ° b =\frac{a+b}{ab}\]Has no identity (i.e. \(a ° b = a\)).

wolf1728
 one year ago
Best ResponseYou've already chosen the best response.0Is that ever true? Let's use some real numbers: 7 * 3 = (7 + 3) / 7*3 7*3 =? 10/21 21 =? 10/21

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Assume that \[a*b=a\] then, \[\frac{ab}{a+b}=a\]\[\frac{1}{a}+\frac{1}{b}=a\]\[\frac{1}{b}=a\frac{1}{a}\]\[\frac{1}{b}=\frac{a^21}{a}\]\[b=\frac{a}{a^21}\] 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

wolf1728
 one year ago
Best ResponseYou've already chosen the best response.0Why do we assume that a*b = a?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0If 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

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0for all a belonging to the required set

ahsome
 one year ago
Best ResponseYou've already chosen the best response.0Ahh, 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?

ahsome
 one year ago
Best ResponseYou've already chosen the best response.0@Nishant_Garg, the \(ab\) should be at the bottom, not the top

ahsome
 one year ago
Best ResponseYou've already chosen the best response.0Yeah, that changes a bit ;)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0but it should be something like that if u just solve it

ahsome
 one year ago
Best ResponseYou've already chosen the best response.0I don't know how to rearrange to get something similar, can you help @Nishant_Garg?

ahsome
 one year ago
Best ResponseYou've already chosen the best response.0Thanks so much @Nishant_Garg :D

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ok 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

ahsome
 one year ago
Best ResponseYou've already chosen the best response.0Maybe, 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?

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0yep, 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...

ahsome
 one year ago
Best ResponseYou've already chosen the best response.0Yeah. Thanks so much @Nishant_Garg :D
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