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Equivalence Relations, Cardinality and Finite Sets.

  1. Dec 5, 2013 #1
    Hey everyone, I have three problems that I'm working on that are review questions for my Math Final.

    1. The problem statement, all variables and given/known data

    First Question: Determine if R is an equivalence relation: R = {(x,y) [itex]\in[/itex] [itex]Z[/itex] x [itex]Z[/itex] | x - y =5}
    and find the equivalence classes.
    Is [itex]Z[/itex] | R a partition?

    2. Relevant equations

    An equivalence class R is reflexive, symmetric and transitive.

    3. The attempt at a solution

    First Question: So I know I have to check reflexivity, symmetricity and transitivity. It seems this relation fails the reflexivity test. Since xRx = {(x,x) -> x-x=5} and x-x ≠ 5. Since it fails, if I'm correct, can I still find equivalence classes and partitions?

    1. The problem statement, all variables and given/known data

    Second Question: Show that |[itex]Z[/itex]+ x [itex]Z[/itex] +| → |[itex]Z[/itex]+| (Where [itex]Z[/itex]+ represents all positive integers)...
    by showing that [itex]Z[/itex]+ x [itex]Z[/itex]+ → [itex]Z[/itex]+ is a bijection.


    2. Relevant equations

    A function is bijective if it is one-to-one and onto.

    3. The attempt at a solution

    Second Question: I know that I have to show that if f(x)=f(y) then x=y but I don't know how to structure it. Also I have to show that f(x) = y [itex]\in[/itex] [itex]Z[/itex]+.

    1. The problem statement, all variables and given/known data

    Third Question: Prove that if n [itex]\in[/itex] [itex]N[/itex] , f:In → B and f is onto then B is finite and |B| < n.

    2. Relevant equations



    3. The attempt at a solution

    Third Question: I don't even know where to begin. I'm not looking for an answer, just more of where to look, where to start..a break down of the concepts addressed in the question would be most helpful.


    Thanks!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Dec 5, 2013 #2

    jbunniii

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    Your example shows that this is not an equivalence relation, because it is not reflexive. So there are no equivalence classes and there is no partition. End of story.
     
  4. Dec 5, 2013 #3

    jbunniii

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    Well, you can't show these properties of ##f## without first defining ##f##. Try to construct a function ##f : \mathbb{Z}^+ \times \mathbb{Z}^+ \rightarrow \mathbb{Z}^+## which is bijective, and then verify that it is bijective. Hint: consider a diagonal argument.
     
  5. Dec 5, 2013 #4

    jbunniii

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    Can you clarify the notation? What does "In" refer to?

    P.S. In the future, it would be better to create separate threads for each question, unless they are closely related.
     
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