Hi, infinite-set readers. Set theory is back.

The previous post we have discussed about the definition of addition on natural numbers. You can visit this link to read the previous post. This time, we will be discuss about the definition of the multiplication of two natural numbers. But before, the theorem will be given as a tool for defining the multiplication of two natural numbers. Happy reading.

**(x) Theorem :**

For any there is a single function such that :

**Proof :**

By taking , , , and is function from to with for all then by Finite Recursion Theorem, this theorem is proven.

The above **Theorem (x)** then becomes a tool for defining the multiplication of natural numbers.

**Definition :**

Given any and are functions defined in the Theorem (x). Defined with .

In the defined of sum and multiplication above (and previous post), it applies associative, commutative, and distributive. But in this post will not be proved these. You can try it yourself to prove the above definition applies these characteristic.

Next will be discussed about the definition of the exponentiation (power) on natural numbers. But before that, will be given the theorem as a tool to define the powers.

**(y) Theorem :**

For any there is a single function such that :

**Proof :**

By take , , , and is function from to wtih foral all then by Finite Recursion Theorem, this theorem is proven.

The **Theorem (y)** above then becomes a tool for defining the exponentiation (power) of natural numbers.

**Definition :**

Given any and are functions defined in the Theorem (y). Defined with .

Furthermore, there will be a correlation between the addition and multiplication operations. As well as the correlation between multiplication operations and forces on natural numbers.

**Theorem :**

For then apply :

(i) as many as

(ii) as many as

**Proof :**

(i) Take any set of natural numbers then :

(ii) Take any set of natural numbers then :

In the defined of power above, it applies associative, commutative, and distributive. But in this post will not be proved these cause my goal is in finite and infinite set. You can try it yourself to prove the above definition applies these characteristic.

See you in my next post. Next, we will begin to discuss finite and infinite sets. Still follow this ‘story’. Thanks for reading. May be useful.

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