This fundamental idea is at the heart of mathematics, because it deals with counting. Just like there is a weight that represents a kilogram, there is a unit in mathematics that represents a number $n$. Let $I(n)$ be the set containing the first $n$ natural numbers. The set $I(n)$ is the “unit” that represents the number $n$. For example, the set $A$ with elements $a, b, c, \dots, z$, has $26$ elements because this set can be put in one to one correspondence with $I(26)$.

But what about infinite sets? The set $N$ of natural numbers is a unit for infinite sets. We say a set $S$ is countable if it can be put in one to one correspondence with $N$. It is remarkable that the following subsets of $N$ are countable. Can you find the one to one correspondence between $N$ and these sets?

1. The set of all even numbers.

2. The set with elements $1, 4, 7, 10, \dots$

3. The set of all rational numbers.

Can you show that the set of prime numbers is countable?

When it comes to infinite sets, a part can be equal to the whole.

1. The set of all even numbers.

2. The set with elements $1, 4, 7, 10, \dots$

3. The set of all rational numbers.

Can you show that the set of prime numbers is countable?

When it comes to infinite sets, a part can be equal to the whole.

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