Pure Mathematics A3: Equivalence Relations

39 flashcards · Mathematics · by lakesare

1. What is the name of the symbol ~? How is A ~ B read?

   ~: tilde A ~ B: A twiddles B

2. Name 3 ways to read 6 ~ 3.

   6 twiddles 3 6 is related to 3 6R3

3. She twiddled the dials on the radio.

   Twist, move, or fiddle with (something), typically in a purposeless or nervous way.

4. When is ~ a relation on set X?

   When: if x, y ∈ X → then x ~ y is either True or False.

5. Give an example of a relation on set ℝ.

   "equal to", e.g. "5 is equal to 5".

6. When is the relation ∼ on a set X an equivalence relation?

   When ~ is: reflexive symmetric transitive

7. Why are these properties considered "independent"?

   Because there exist relations with every combination of these properties.

8. Why is this non-mathematical relation on people is an equivalence relation?

   Because it's: symmetric, reflexive and transitive.

9. Name the 2 meanings the word "parallel" can take in mathematical texts.

   parallel - 2 lines never meet parallel - "in the same direction as"

10. What do the first 3 properties of congruences imply?

    That congruence modulo n is always an equivalence relation on ℤ.

11. Express the "congruence modulo n relation" in this form.

    The relation ~ defined on ℤ by a ~ b if (a - b) = nk, also expressed as: a ≡ b (mod n)

12. How can you roughly think about equivalence relations?

    Very roughly, you can think of any equivalence relation as a relation that defines some kind of "equivalence" on the objects in the set on which the relation is defined.

13. In what sense are numbers equivalent with the equivalence relation "is equal to" on the set R?

    a ~ b if the numbers are equal, even if they are written differently.

14. In what sense are lines equivalent with the equivalence relation "is parallel to" on the set of lines on a plane?

    a ~ b if these lines have the same direction, and we're only interested in their direction.

15. In what sense are lines equivalent with the equivalence relation "congruence modulo n" on the set of Z?

    a ~ b if they have the same reminder on division by n.

16. Relation VS equivalence relation?

    relation equivalence relation

17. What's equivalence class of x? Given ~ is the equivalence relation on a set X, and x ∈ X.

    The set of all elements in X that are equivalent to x.

18. What does [[x]] denote?

    The equivalence class of x. This idea is associated only with equivalence relations, not with relations in general.

19. Consider the equivalence relation "was born in the same year as" on a set of people. What's the equivalence class of a particular person?

    The equivalence class of a particular person is the set of people who were born in the same year as that person, including the person themself.

20. What can you say about [[l]] and [[m]]?

    [[l]] and [[m]] are the same set.

21. What's an good way to imagine that "equivalence classes partition X"?

22. How would you prove this?

23. Say in words what's [[0]] here?

    The equivalence class of 0 is the subset of Z containing all the integers that are congruent to 0 modulo 5.

24. What's this relation in a function form? Every relation can be seen as a function.

    ~: Z×Z → { True, False } (x, y) → x - y is an integer

25. Given this relation, how can we describe its equivalence class [[a]]? ~: Z×Z → { True, False } (x, y) → x - y is an integer This relation is known to be the equivalence relation.

    All such y that "a - y is an integer". The order doesn't matter, because this relation is reflexive!

26. How did we jump from –2k to 2k?

    These sets of real numbers are the same.

27. What's a nice way to write down the solution to this?

28. Tilde VS Tilda?

    Tilde - ~ The tilde, ˜ or ~, is a grapheme with several uses. The name of the character came into English from Spanish and Portuguese, which in turn came from the Latin titulus, meaning "title" or "superscription".Tilda Swinton - short form of Matilda. Tilda Swinton

29. Tweed VS twiddles?

    tweed - rough fabric Tweed is a rought, woolen fabric of a soft, open, flexible texture, resembling cheviot or homespun, but more closely woven.to twiddle - twist, move, or fiddle with (something), typically in a purposeless or nervous way.

30. Name 2 steps to find all equivalence classes of a relation.

    Find [[0]], [[1]], etc. Make sure these equivalence classes make a partition. [[0]] ∪ [[x]] = ℤ

31. How else can we denote [[1]]? The equivalence classes of the equivalence relation "congruence modulo 5".

    [[–4]], [[6]], etc.

32. What do we call [[1]] if we choose it to denote its equivalence class? The equivalence classes of the equivalence relation "congruence modulo 5".

    Representative of the equivalence class.

33. What's the set of representatives for the equivalence relation "congruence modulo 5"?

    { 0, 1, 2, 3, 4 }

34. How would you solve 3 +5 4 using this definition?

35. Describe the set of representatives visually.

    A set of representatives is [0, ∞)

36. Brick and __________?

    Mortar Строительный раствор — раствор из вяжущего вещества, воды и заполнителя (возможны добавки), со временем превращающийся в искусственный камень.

37. What's equivalence class [[x]] in simple words?

    All elements of the domain that are the same as x (in our perverted sense of sameness).

38. Describe the formula for "congruence modulo 2π".

    ~: R×R → { True, False } (x, y) → x - y = 2πk for some k∈ℤ

39. Why do we like to define our representatives in the relation "congruence modulo 2π" as (–π, π]?

    Because it corresponds to the definition of the principal argument of a complex number.