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Hey everyone, I have three problems that I'm working on that are review questions for my Math Final.
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?
An equivalence class R is reflexive, symmetric and transitive.
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?
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.
A function is bijective if it is one-to-one and onto.
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]+.
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.
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!
Homework Statement
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?
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 |[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.
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 [itex]\in[/itex] [itex]Z[/itex]+.
Homework Statement
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.
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!