Math Proofs

A logical argument supported with established facts, following a logical sequence to establish the truth of a Lemma/Theorem.

1. You have a $F$ Theorem that you want to prove or disprove Rigourously
2. You work through steps to prove it

Statement Types

$P⟹Q$

First assume $P$ is true, then show $Q$ is also true We call $P$ the hypothesis, and $Q$ the conclusion.

$\forall x\in D,P(x)$

• for all objects of a given type First let $x$ be an arbitrary element of D, then prove $P(x)$ is true.

$\exists x\in D$ such that $P(x)$

• For at least 1 object of a given type
  First choose $x$, then prove both $x\in D$ and $P(x)$ are true.

Proof Types

Proving

• Proof By Direct Proof
• Proof By Contra-Positive
• Proof By Contradiction
• Proof By Counter Example
• Proof By Induction

Subtypes

• Exhaustive Proof

’Real Proofs’

https://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Challen/proof/proof.html

Proofs

• 2 Odd Integers Add To Even Integer
• Euclid Infinitely Many Prime Numbers
• Square Has Same Parity Proof
• Square Root Is Irrational Proof
• Square Root Function Increasing Proof

Philosophy

• Mathematical Communication

Properties, Laws, Axioms, Etc.

• Triangle Inequality
• Reverse Triangle Inequality
• Inequality
• Interval
• Real Number
• Integer
• Quadratic Formula