# Using cardinatility of sets in discrete mathematics the value of N is real numbers Let A be a collection of sets such that X\in A if and only if X\subset N and |X|=n for some n\in N Prove that |A|=|N|.

Using cardinatility of sets in discrete mathematics the value of N is real numbers
Currently using elements of discrete mathematics by Richard Hammack chapter 18
Let A be a collection of sets such that $$\displaystyle{X}\in{A}$$ if and only if $$\displaystyle{X}\subset{N}$$ and $$\displaystyle{\left|{X}\right|}={n}$$ for some $$\displaystyle{n}\in{N}$$
Prove that $$\displaystyle{\left|{A}\right|}={\left|{N}\right|}.$$

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Step 1
Given that, A is collection of sets such that $$\displaystyle{x}\in{A}$$ if and only if $$\displaystyle{X}\subset{N}$$ and $$\displaystyle{\left|{X}\right|}={n}$$ for some $$\displaystyle{n}\in{N}$$
Here, it means that A is collection of finite subset of N
Now need to show $$\displaystyle{\left|{A}\right|}={\left|{N}\right|}.$$
It is sufficient to show A is countable.
Step 2
Use induction method to show A is countable.
a: The number of subsets of N with cardinality 1 is finite. It is trivially true.
b: Assume that, the number of subsets of N with cardinality K is finite.
Step 3
c: Need to show the number of subsets of N with cardinality $$\displaystyle{K}+{1}$$ is finite.
The number of subsets of N with cardinality $$\displaystyle{K}+{1}$$ is exactly rwo times of number of subsets of cardinality K as each subset has two choices either $$\displaystyle{\left({K}+{1}\right)}^{{{t}{h}}}$$ term belongs to the set or not.
Thus, the number of subsets of N with cardinality $$\displaystyle{K}+{1}$$ is finite.
Hence, for all $$\displaystyle{n}\in{N}$$ the number of subsets of N with cardinality n is finite.
Therefore, $$\displaystyle{A}={\bigcup_{{{n}={1}}}^{{\infty}}}{A}_{{{n}}}$$
Step 4
It is known that, countable union of countable set is countable.
Here, $$\displaystyle{A}_{{{n}}}$$ is countable for each $$\displaystyle{n}\in{N}$$
So, A is countable.
Hence, $$\displaystyle{\left|{A}\right|}={\left|{N}\right|}$$ is proved.