Contradiction Proof Technique

Contradiction Proof Technique
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The Contradiction Proof Technique is one of the common proof techniques used when the Forward-Backward Proof Technique is not suitable for the given proof problem.
Suppose If A then B is a proposition involving statements A and B. The contradiction proof technique (Figure 121.3) begins by assuming that B is false (i.e, not B). The problem is proved if at the end the assumption is contradicted.

(a) Prove, by contradiction, that if n is an integer and n² is even, then n is even.
(b) Prove, by contradiction, that at a party of x people, where x ≥ 2, there are at least two people who have the same number of friends at the party.
(c) Prove, by contradiction, that there are an infinite number of primes.

The strings: S₇P₂A₂₁ (Identity - Physical Property).

The math:
Pj Problem of Interest is of type identity (physical property). Proofs establish truths. So they are identity problems.

Contradiction Proof Technique

(a) Initial assumption: not B.
So, for n² even, n is odd.
So, there is an integer k such that n = 2k + 1.
So, n² = (2k + 1)²
So, n² = 4k² + 4k + 1
So, n² = 2(2k² + 2k) + 1
So, n² is odd.
So, initial assumption is contradicted.

(b) Initial assumption: not B.
So, no two people at the party has the same number of friends
So, the people at the party can be numbered in such away that can be represented by an ordered pair (p,f). Where p is the xth person at the party and f is the number of friends p has. The ordered pair will look like this:
(1,0), (2,1) ...(x, x-1)
It then follows that the person with (x-1) friends is a friend with the person with no friends. A contradiction.
So, initial assumption is contradicted.

(c) Initial assumption: not B.
So, there are a finite number of primes.
Let n be the largest prime
Let k be any prime divisor of n! + 1.
So, n ≥ k, since n is the largest prime.
But n! + 1 cannot be divided by a number between 1 and n
So, k > n. A contradiction.
So, initial assumption is contradicted.

Math

The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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Blessed are they that have not seen, and yet have believed. John 20:29