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Zaara

  • 3 years ago

Group Theory: any one pls guide me with a link to find the definition of these......... 1.(Zm,+) 2.(Zm,*)

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  1. experimentX
    • 3 years ago
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    what is Zm?

  2. Zarkon
    • 3 years ago
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    they prob mean \(Z_m=\{0,1,2,\ldots,m-1\}\)

  3. Zarkon
    • 3 years ago
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    the set of integers mod \(m\)

  4. Zaara
    • 3 years ago
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    Yeah!!!! i need complete sentences to explain it... do u have any stuff???

  5. experimentX
    • 3 years ago
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    show that it satisfies the property to be called a group.

  6. Zaara
    • 3 years ago
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    Exactly!!! can i get that proof or its details... i mean any link???

  7. experimentX
    • 3 years ago
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    let 'a' and 'b' be two elements of Zm, show that it is closed in +, ie, \( (a+b) \mod b \in Z_m \)

  8. experimentX
    • 3 years ago
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    addition operation is associative, and find the identity element of Zm

  9. experimentX
    • 3 years ago
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    Edit:: \[ (a+b) \mod m \in Z_m \]

  10. experimentX
    • 3 years ago
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    this is always true becuse \( (a+b) \mod m = c < m, c \in Z\m \)

  11. Zaara
    • 3 years ago
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    What i have written below is correct right?? or should there be any change????? Definitions: Let Zm be the set of non negative integers less than m: {0,1, ., m−1} The operation +m is defined as a +m b = (a + b) mod m. This is addition modulo m. The operation ∙m is defined as a ∙m b = (a + b) mod m. This is multiplication modulo m. Using these operations is said to be doing arithmetic modulo m. Example: Find 7 +11 9 and 7 ·11 9. Solution: Using the definitions above: – 7 +11 9 = (7 + 9) mod11 = 16 mod11 = 5 – 7 ·11 9 = (7 ∙ 9) mod11 = 63 mod 11 = 8

  12. experimentX
    • 3 years ago
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    huh!! what exactly is the question?

  13. Zaara
    • 3 years ago
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    I wanna define it with examples!

  14. experimentX
    • 3 years ago
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    aren't those two different distinct groups?

  15. Zaara
    • 3 years ago
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    u mean the addition modulo and the other???

  16. experimentX
    • 3 years ago
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    yep!! just this one .. (Zm,+), this is a group.

  17. Zaara
    • 3 years ago
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    Thats it...what is confusing u??

  18. experimentX
    • 3 years ago
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    you are mixing two things ... doing both of them at same time.

  19. Zaara
    • 3 years ago
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    where did i mix them?? i have defined Zm,then defined addition and multiplication modulo and have given example 4 each...

  20. experimentX
    • 3 years ago
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    let's work out one at each ... we do with (Zm,+) first.

  21. Zaara
    • 3 years ago
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    ohhh... u mean i have not seperately done it..ryt??? ok got it.... i have jst given it together for my ease....

  22. experimentX
    • 3 years ago
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    but why do you want to give example?

  23. Zaara
    • 3 years ago
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    definition with example is most expected... so that one can easily understand it....

  24. experimentX
    • 3 years ago
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    choose m=10, or 5 for sake of ease. then {0,1,2,3,4} are the elements of group!! show that it remains closed, and is associative and has identity '0', and has inverse.

  25. Zaara
    • 3 years ago
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    then is my answer wrong????

  26. experimentX
    • 3 years ago
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    no i don't think so ... you put stuff in one place, and it hurt my eyes.

  27. Zaara
    • 3 years ago
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    is it ok now??? Let Zm be the set of nonnegative integers less than m: {0,1, ., m−1} 1)The operation +m is defined as a +m b = (a + b) mod m. This is addition modulo m.(Zm,+) Example: Find 7 +11 9 Solution: Using the definitions above: 7 +11 9 = (7 + 9) mod11 = 16 mod11 = 5 2)The operation ∙m is defined as a ∙m b = (a + b) mod m. This is multiplication modulo m. Example: Find 7 ·11 9. Solution: Using the definitions above: 7 ·11 9 = (7 ∙ 9) mod11 = 63 mod 11 = 8

  28. Zaara
    • 3 years ago
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    @experimentX

  29. experimentX
    • 3 years ago
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    sorry ... i was busy!!

  30. Zaara
    • 3 years ago
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    if possible make some alterations and input the needed stuffs... and edit my answer...pls:)

  31. experimentX
    • 3 years ago
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    okay .. you showed that it is closed. you need to test for these http://en.wikipedia.org/wiki/Group_(mathematics)#Definition

  32. Zaara
    • 3 years ago
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    shud i show associativity,identity element and inverse element????

  33. Zaara
    • 3 years ago
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    @experimentX

  34. experimentX
    • 3 years ago
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    yes!! you need to show them!!

  35. Zaara
    • 3 years ago
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    bt in the link u gave... they have shown for multiplication modulo only.... what can i do for addition modulo???????

  36. experimentX
    • 3 years ago
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    same ... show that (a+b) mod m is in the set!!.

  37. experimentX
    • 3 years ago
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    instead of multiplying, we add ,,, since operation is +.

  38. terenzreignz
    • 3 years ago
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    Never seen a Group Theory question before... this should be fun... :D

  39. terenzreignz
    • 3 years ago
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    Once you've shown closure, next step is to show that there is an element e of Zm (a non-negative integer less than m) such that for any k in Zm e + k = k + e = k

  40. Zaara
    • 3 years ago
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    similar to that??? r u sure???

  41. terenzreignz
    • 3 years ago
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    + here is + modulo m, ok

  42. terenzreignz
    • 3 years ago
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    Quite sure.

  43. terenzreignz
    • 3 years ago
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    Okay, I'll do this (because it's easier :P) and you do the "existence of inverse" part. Let k be in Zm Consider 0. 0 is a non-negative integer. Since m is positive, 0 < m So 0 is in Zm 0 + k = 0 + k (mod m) = k(mod m) = k, since k is in Zm (and is therefore a nonnegative integer less than m). Similarly, k + 0 = k + 0 (mod m) = k(mod m) = k. Thus, we have shown that 0 + k = k + 0 = k Hence, 0 is the identity element in <Zm, +>.

  44. Zaara
    • 3 years ago
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    what abt inverse???

  45. terenzreignz
    • 3 years ago
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    -.- Here's the gist of it. Consider an arbitrary element of Zm, and show that there is something that you can add (modulo m) to it so that the sum is 0 (the additive identity).

  46. experimentX
    • 3 years ago
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    show that for a, m-a is the it's inverse, and m-a is also in the set.

  47. terenzreignz
    • 3 years ago
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    @experimentX A spoiler alert, next time? LOL JK

  48. experimentX
    • 3 years ago
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    well .. sorry!!

  49. Zaara
    • 3 years ago
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    why???

  50. terenzreignz
    • 3 years ago
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    Never mind, lol... Anyway, let a be in Zm. Your task is to show that an element m-a exists, and m-a is in Zm, and a + (m - a) = 0, where "+" here is + (mod m)

  51. Zaara
    • 3 years ago
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    why m-a??? i dnt get it...

  52. terenzreignz
    • 3 years ago
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    Well, remember that in the world of Zm, every multiple of m is basically zero. So, when I said "what do you need to add to a so that you get zero", I really mean what do you need to add to a to get m...

  53. terenzreignz
    • 3 years ago
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    Okay, first of all, the inverse of the additive identity is itself... 0 + 0 = 0, so, yeah... Now, let a be in Zm, where a is not equal to zero. Consider m-a. since a is in Zm, a < m a - a < m - a m - a > 0, meaning m-a is non-negative, (positive, even) m < m + a m - a < m + a - a m - a < m, meaning m-a is less than m. Thus, m-a is non-negative and is less than m, therefore m-a is in Zm. Consider a + (m-a) = a + m - a = m = 0(mod m) Thus, a + (m-a) = 0(mod m) a + (m-a) = 0, in Zm (m-a) is the inverse of a. Thus we have shown that for any element a in Zm, there is an element (m-a) such that a + (m-a) = 0, and therefore, all elements in <Zm, +> have inverses :D

  54. terenzreignz
    • 3 years ago
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    And there you go, we've shown closure, associativity, existence of an identity element and existence of inverses, we can conclude that <Zm, +(mod m)> forms a group. (An Abelian group, too, but more on some other time, I guess)

  55. Zaara
    • 3 years ago
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    great!!!!!!!!!!!!!!

  56. terenzreignz
    • 3 years ago
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    I have to go now, work on the other one: <Zm, *(mod m)> But since we're having spoilers anyway, it's not a group :P The reason for that is for you to find... *cough*oneisthemultiplicativeidentity*cough*theresnothingyoucanmultiplytozerotogetone *cough*zerohasnomultiplicativeinverse *cough* LOL anyway Terence out :D

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