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Now you are moving into material and proofs that really do depend upon the particular set of axioms and theorems given in your text material.
How does your text define finite sets & infinite sets? There are two basic approaches. Which does your text use?
Give us some idea about the basic theorems you have to use.
Sorry but what you posted tells us nothing. If you want some help with these topics you must answer the following two questions.
1) How does your text define finite sets & infinite sets? There are two basic approaches. Which does your text use?
2) What are the basic theorems you have to use?
This is by far the more difficult of the two approaches. So what I do may not suit your text/instructor.
Here is a very important theorem: There is a injection from X to Y if and only in there is a surjection from Y to X.
We know that X is not finite, so $\displaystyle \left| X \right| \ge \aleph _0 $. But because there is a injection from X to N then there is a surjection from N to X. But on order for that to be true we must have $\displaystyle \aleph _0 = \left| N \right| \ge \left| X \right|$. From these two we see $\displaystyle \aleph _0 = \left| X \right|$.
Given that Y is not infinite then, how would you show that Y bijects with some natural number? Here I don’t know what theorems you have to use.