离散数学——Mathematical Logic

目录

• Propositional Logic
• Predicate Logic
• Rules of Inference
  • For Propositional Logic
  • For Quantified Statements
• Proving Method

Propositional Logic

A proposition is a declarative statement that is either true or false. More complex propositions can be built from elementary statements using logical connectives.

Two propositions are equivalent if they always have the same truth value.

The propositions p and q are called logically equivalent if p ↔ q is a tautology, denoted by p ≡ q or p ⇔ q.

Predicate Logic

A predicate P(x) assigns a value T or F to each x depending on whether the property holds or not for x

Two types of quantified statements: universal and existential

The truth values of ∃x P(x) and ∀x P(x) depend on both the propositional function P(x) and the universe.

The order of nested quantifiers matters if quantifiers are of different type

The order of nested quantifiers does no matter if quantifiers are of the same type.

nested negation

Rules of Inference

For Propositional Logic

For Quantified Statements

Proving Method