Prime number, definition by sets

PRIME NUMBER SERIES 3

Prime number, definition by sets

A number p is prime, if

p ∈ (ℕ \ {0,1}) \ (ℕ \ {0,1})²

Proof:

ℕ = {0,1,2,…}

ℕ \ {0,1} = {2,3,4,…}

(ℕ \ {0,1})² = (ℕ \ {0,1}) x (ℕ \ {0,1}) = {2,3,4,…} x {2,3,4,…} = {{4,6,8,…},{6,9,12,…},{8,12,16,…},…}

p ∈ {2,3,4,…} \ {{4,6,8,…},{6,9,12,…},{8,12,16,…},…}

Lessening set {2,3,4,…} by sets {{4,6,8,…},{6,9,12,…},{8,12,16,…},…}

results in

p ∈ {2,3,5,7,11,13,17,19…}

The set of the natural numbers comprise of

the numbers 0 and 1,

composite numbers, which are the product of at least two natural numbers (both greater than 1)

as well as all none composite numbers greater than 1, which are not the product of two natural numbers (both greater than 1). And those can be only prime numbers.

There do not exist any other numbers in the set of natural numbers.

Because of this

p ∈ (ℕ \ {0,1}) \ (ℕ \ {0,1})²

is true.

What had to be proved.

Munich, 18 October 2022

Gottfried Färberböck