Mathematical Statement

Abstract

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• Also known as Proposition
• Can be presented with variables like p, q, r & s which return either True or False etc. Statements expressed in variables are called Statement Form which isn’t mathematical statement because we can’t tell if it is True or False
• 3 important types are Conditional Statement, Universal Statement, & Existential Statement. We can form complex statements that are made of more than one type

Keep It Atomic

This makes the cognitive load low, easier to build on top of the statement, especially for Mathematical Proof that is complicated

Attention

1. Predicate without value substituted to its Predicate Variable isn’t mathematical statement, unless it is Logical Equivalence equation that involves predicate. For example $\neg (\forall x,P(x))\equiv \exists x,\neg P(x)$
2. Not in a question form
3. Either True or False, but not both at the same time

Compound Statement

• Make up of multiple Mathematical Statement, connected with Logical Connectives

Simplification

When the statement has Conditional Statement, convert it using Implication Law to make it much less confusing

Special Mathematical Statements

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Axioms (公理)

• Universally recognized truth or principle
• Mathematical Statement assumed to be true without Mathematical Proof

Theorem (定理)

• Mathematical Statement that is proved using Mathematical Reasoning
• Usually a major or important result

Lemma (引理)

• A small Theorem (定理) used to help in proving a bigger theorem

Corollary (推论)

• A result that is a simple deduction from a Theorem (定理)
• For example, we proof that the product of any two odd numbers is always odd. The corollary is the product of two sonsecutive odd numbers is always odd

Conjecture (猜测)

• Mathematical Statement believed to be true, but there isn’t a Mathematical Proof yet

Terminologies

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Tautology

• Mathematical Statement that is always true

Contradiction c

• Mathematical Statement that is always false