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Ahsome

  • one year ago

A simple * question

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  1. anonymous
    • one year ago
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    whats the question

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

  3. ahsome
    • one year ago
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    Any clue?

  4. ahsome
    • one year ago
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    Here is what I got

  5. wolf1728
    • one year ago
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    Is that ever true? Let's use some real numbers: 7 * 3 = (7 + 3) / 7*3 7*3 =? 10/21 21 =? 10/21

  6. anonymous
    • one year ago
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    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

  7. wolf1728
    • one year ago
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    Why do we assume that a*b = a?

  8. anonymous
    • one year ago
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    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

  9. anonymous
    • one year ago
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    for all a belonging to the required set

  10. ahsome
    • one year ago
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    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?

  11. ahsome
    • one year ago
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    @Nishant_Garg, the \(ab\) should be at the bottom, not the top

  12. anonymous
    • one year ago
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    oh wow my bad haha

  13. ahsome
    • one year ago
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    Yeah, that changes a bit ;)

  14. anonymous
    • one year ago
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    but it should be something like that if u just solve it

  15. ahsome
    • one year ago
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    I don't know how to rearrange to get something similar, can you help @Nishant_Garg?

  16. anonymous
    • one year ago
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    ill try

  17. ahsome
    • one year ago
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    Thanks so much @Nishant_Garg :D

  18. anonymous
    • one year ago
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    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

  19. ahsome
    • one year ago
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    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?

  20. anonymous
    • one year ago
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    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...

  21. ahsome
    • one year ago
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    Yeah. Thanks so much @Nishant_Garg :D

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