# Thread: Proof - A is Infinite

1. ## Proof - A is Infinite

I know that to prove A is finite, I say:

$\displaystyle \exists$ a bijection $\displaystyle f : N_k \rightarrow A$

So to prove A is infinite would I say:

$\displaystyle \exists$ a bijection $\displaystyle f : \mathbb{N} \rightarrow A$

Our professor just said "not finite." So I am a little confused as to what to negate.

2. Originally Posted by Pi R Squared
I know that to prove A is finite, I say:
$\displaystyle \exists$ a bijection $\displaystyle f : N_k \rightarrow A$

So to prove A is infinite would I say:
Show that $\displaystyle (\forall~f)(\forall~k)[f : N_k \rightarrow A]$ then $\displaystyle f$ is not a bijection.

That is $\displaystyle A$ is infininte if it is not finite.
$\displaystyle A$ does not biject with any $\displaystyle N_k$.

3. ## Proof - A is Infinite

Originally Posted by Plato
Show that $\displaystyle (\forall~f)(\forall~k)[f : N_k \rightarrow A]$ then $\displaystyle f$ is not a bijection.

That is $\displaystyle A$ is infininte if it is not finite.
$\displaystyle A$ does not biject with any $\displaystyle N_k$.

I am understanding this a little, but still not enough to prove it. This is the problem I am working on:

Proof: Suppose for the sake of obtaining a contradiction that if C is an infinite set then D is not an infinite set or $\displaystyle g \rightarrow C$ is not one-to-one.

I clearly see the logic. But am confused as to how to relate $\displaystyle [f : N_k \rightarrow C]$ to $\displaystyle g \rightarrow C$.

I guess I am confused as to how $\displaystyle [ N_k ]$ can be infinite...

4. Originally Posted by Pi R Squared
I guess I am confused as to how $\displaystyle [ N_k ]$ can be infinite...
You really need to get this clear in your brain.
There very basis of this approach to finite set is that $\displaystyle \left( {\forall j \in \mathbb{Z}^ + } \right)$ the set $\displaystyle N_j=\{1,2,\cdots,j\}$ is finite.
That means that by definition $\displaystyle N_k$ cannot be infinite.

5. ## I understand...

Originally Posted by Plato
You really need to get this clear in your brain.
There very basis of this approach to finite set is that $\displaystyle \left( {\forall j \in \mathbb{Z}^ + } \right)$ the set $\displaystyle N_j=\{1,2,\cdots,j\}$ is finite.
That means that by definition $\displaystyle N_k$ cannot be infinite.

I understand...

[tex]N_k[\math] is finite.

So I am going to have to prove that if C is an infinite set then D is not, therefore proving by contradiction that the statement is true.

I wonder if you could clarify this statement using onto and/or 1-1.

Originally Posted by Plato
does not biject with any .