A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

jango_IN_DTOWN

  • one year ago

Equivalence relation problem

  • This Question is Closed
  1. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    1 Attachment
  2. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @ganeshie8

  3. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    |dw:1444584458933:dw|

  4. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    for each relation, we need to check : 1) reflexive 2) symmetry 3) transitive

  5. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    for i : reflexive : \((x_1, y_1)\sim (x_1,y_1)\) because \(y_1=y_1\) symmetry : \( (x_1,y_1)\sim(x_2,y_2) \implies (x_2,y_2)\sim (x_1,y_1)\) because \(y_1=y_2 \implies y_2=y_1\) transitivity : \( (x_1,y_1)\sim(x_2,y_2) \) and \((x_2,y_2)\sim (x_3,y_3)\) \(\implies (x_1,y_1)\sim (x_3,y_3)\) because \(y_1=y_2\) and \(y_2=y_3\)\(\implies y_1=y_3\) therefore, this is an equivalence relation

  6. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    the reflexive part in i) I am having a confusion

  7. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    to better understand reflexivity, maybe consider a relation that is not reflexive

  8. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    first of all, as the name says, "reflexive" refers to the reflection that you see when u look at mirror. you see your own reflection... we say a relation is reflexive if \(x\sim x\) for all \(x\) in the relation.

  9. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    can you think of a relation that is not reflexive ?

  10. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes a>b

  11. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    nice, another one : a-b is odd

  12. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yeah

  13. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so do you get why the relation in part i is reflexive ?

  14. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes now it is clear..

  15. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    review quick top 3 properties |dw:1444585876121:dw|

  16. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    checked..

  17. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    can you guess what the equivalence classes will be

  18. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    x will be any member of R and y will be fixed

  19. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Exactly! for example below is an equivalence class : [(x, 1)] = {(1,1), (2,1), (2.2, 1), (-99, 1), ... }

  20. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    below is another equivalence class [(x, 3)] = {(1,3), (2,3), (2.2, 3), (-99, 3), ... }

  21. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    [(x,y)]={(a,b)belongs to R^2 such that b=y}

  22. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    looks nice

  23. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so this is the answer right?

  24. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Yes thats the answer for part i

  25. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    look at the relation in part ii, whats ur first guess, can it be an equivalence relation ?

  26. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    I think it is reflexive and symmetric but transitivity cant say

  27. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    right, just show an example that its not transitive

  28. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    then we need to consider general points of R^2

  29. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes just pick any one simple example

  30. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    (1,2)~(1,3) and(1,3)~(2,3) but (1,2) is not ~ to (2,3)

  31. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    that will do, that proves the relation is not transitive consequently its not equivalence relation

  32. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes correct and hence no question of equivalence classes arise

  33. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    good iii looks innocent, but it can be very tricky...

  34. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    because there are several ways to get an integer by taking difference of two numbers : 3 - 1 = integer 1.4 - 0.4 = integer 0.3 - 0.3 = integer ..

  35. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    ohhhh

  36. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    proving that it is an equivalence relation is trivial but figuring out equivalence classes can be tricky...

  37. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    we see that the y component can be anything

  38. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    hey wait, does it really pass transitivity ?

  39. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yeah if a-b is an integer and b-c is an integer then a-c must be an integer

  40. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    a-c=(a-b)+(b-c)= sum of integers

  41. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Ahh thats clever! okay so it does pass transitivity

  42. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yes.. and the symmetric and reflexive parts are also satisfied.. so we need to figure out the equivalence realtion

  43. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    you mean equivalence `classes`

  44. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yeah oops

  45. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    so our y component can be anything

  46. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    but x component will be those real numbers whose difference gives integer

  47. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    How about this [(x,y)]={(a,b)belongs to R^2 such that a = x-floor(x)+k, \(k \in \mathbb Z\)}

  48. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    floor(x) gives the integer part of x so, x - floor(x) gives the fractional part of x

  49. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yeah it gives integer always I thought of another one, [(x,y)]={(a,b)belongs to R^2 such that (x-a) belongs to Z}

  50. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    looks much better!

  51. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    But both will work

  52. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    mine is like construction your's is more like a logic statement yeah both works, but yours looks better

  53. jango_IN_DTOWN
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Thanks. so we are done with the questions. I just need to construct the language only..:)

  54. ganeshie8
    • one year ago
    Best Response
    You've already chosen the best response.
    Medals 1

    np :)

  55. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.