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Equivalences (me trying to understand an example)

  1. Jun 10, 2014 #1
    1. The problem statement, all variables and given/known data
    I'll be using ≈ as the equivalence symbol.

    If m and n are integers, define m≈n to mean that m-n is even. Then ≈ is an equivalence on Z.

    [0] = {x→Z | x≈0} is the set of even integers
    [1] = {x→Z | x≈1} is the set of odd integers

    where the x→Z was used to represent X is an element of Z.

    So basically i'm having trouble making sense of this...
    x≈0 means x-0, and if that has an equivalenet class of [0] somehow that means it's even? Can somebody try to explain what's going on here to me?
  2. jcsd
  3. Jun 10, 2014 #2


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    [0] and [1] are notations for equivalence classes. All elements of, say, [0] are related to 0 via the equivalence relation. This means that [0], for example, contains all integers, x, such that x ≈ 0. What does x≈0 mean? It means that for any x in the integers, x - 0 = x is even. Well this is simply the set of all even integers.

    Does [1] now make sense?
  4. Jun 10, 2014 #3


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    Here, have an \in symbol: ∈

    If you're going to use plain text, you can copy and paste that symbol instead of that rather confusing arrow. You can also use LaTeX. Type
    Code (Text):
    ##[0]=\{x\in\mathbb Z|x\approx 0\}##
    to get ##[0]=\{x\in\mathbb Z|x\approx 0\}##. See the FAQ post for more information.
  5. Jun 10, 2014 #4


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    It means that x-0 is an even number and since x- 0= x, that is the same as saying x os an even number.

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