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LOGIC AND REASONING, MATHEMATICAL PROOFS - Coggle Diagram

- - - - Negation: p vs. -p
- - - - implication and its logical equivalent
      - The converse and inverse of an implication
      - Double Negation
      - Equivalence Laws
        
        Commutative
        
        Idempotent
        
        De Morgan's
        
        Associative
        
        Disributive
- - - - Proposition p
      - Proposition q
      - T or F
    - - Conjunctions: ^
      - Disjunction: v
      - Xor: ⊕
      - Implication: →
      - Biconditional: ⇔
- - - - -p -> -q
    - - -q -> -p
    - - q -> p
- - - - Universal Quantification
        
        ALL
      - Existential Quantification
        
        SOME

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LOGIC AND REASONING

Truth tables

Propositions

Truth Values

Declarative & Complete

Either not Both

True

False

True denoted by T

Negation: p vs. -p

False denoted by F

Binary Propositional Connectives

Conditional Propositions

Truth tables

Truth value

Symbols

Conjunctions: ^

Disjunction: v

Xor: ⊕

p -> q

Implication: →

Inverse

Contrapositive

Biconditional: ⇔

Converse

q -> p

Proposition p

-p -> -q

Proposition q

-q -> -p

T or F

Universe of Discourse

Universal set U

Quantifiers

Universal Quantification

Existential Quantification

Compound proposition

Tautology

Contradiction

Contingency

Logical Equivalence (if p⇔q is a Tautology)

Always False

Always True

Opposite of Tautology

Neither Tautology or Contradiction

Varying truth values

ALL

SOME

implication and its logical equivalent

The converse and inverse of an implication

Double Negation

Equivalence Laws

Commutative

Idempotent

De Morgan's

Associative

Disributive

MATHEMATICAL PROOFS

Types of Reasonings

Deductive

general to specific

Fallacies

Inductive

Affirming the conclusion

specific to general

Begging the question or Circular reasoning

Rules of Inference

Addition

Simplification

Conjunction

Modus ponens

Modus tollens

Hypothetical syllogism

Disjunctive syllogism

Denying the hypothesis

Methods of proving

Vacuous

Direct

Indirect

Trivial

MATH AS A LANGUAGE

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