Equivalence Relations, Cardinality and Finite Sets.

[3=MCsq]

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

Homework Statement

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

Homework Equations

An equivalence class R is reflexive, symmetric and transitive.

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?

Homework Statement

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

Homework Equations

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

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 \in Z⁺.

Homework Statement

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

Homework Equations

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!

 

[jbunniii]

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?

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.

 

[jbunniii]

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 \in Z⁺.

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.

 

[jbunniii]

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

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.