Week 9 - Conditional Statements (Vocabulary)

Vocabulary terms from Week 9 notes on conditional statements, including identification of statements, negation, quantifiers, symbolic logic, truth tables, De Morgan's laws, and how conjunctions/disjunctions affect truth values.

Statement

A declarative sentence that is either true or false (but not both).

Simple statement

A statement with a single proposition and no logical connectives.

Compound statement

A statement formed by combining two or more simple statements using logical connectives (e.g., and, or).

Negation

The logical operation that reverses the truth value of a statement, typically denoted by ~.

Quantifier

A word that indicates how many elements in a domain satisfy a property (e.g., all, some).

Universal quantifier

Indicates that every element in the domain satisfies the property (often expressed as 'for all').

Existential quantifier

Indicates that at least one element satisfies the property (often 'there exists' or 'some').

Negation of All S are P

The negation of 'All S are P' is 'There exists an S that is not P'.

Negation of Some S are P

The negation of 'There exists an S that is P' is 'No S is P' (equivalently, 'All S are not P').

All vs Some (quantifiers)

All = universal quantification; Some = existential quantification.

Symbolic translation

Expressing statements with symbols (e.g., p, q, r, ~, ∧, ∨).

Negation symbol (~)

The symbol used to negate a statement (NOT).

Conjunction

The ∧ operator meaning 'and' — true when both components are true.

Disjunction

The ∨ operator meaning 'or' (inclusive) — true when at least one component is true.

Truth table

A chart listing all possible truth values for propositions and the resulting truth value of a compound statement.

Propositional variables

Letters like p, q, r used to stand for simple propositions in truth tables.

De Morgan's Laws

Rules: not (P ∧ Q) ≡ (¬P ∨ ¬Q) and not (P ∨ Q) ≡ (¬P ∧ ¬Q).

Negation of a conjunction

Negating P ∧ Q yields ¬P ∨ ¬Q (per De Morgan's Law).

Negation of a disjunction

Negating P ∨ Q yields ¬P ∧ ¬Q (per De Morgan's Law).

Testing a claim with logic (conjunction vs disjunction)

A conjunction claim is true only if all parts are true; a disjunction claim is true if at least one part is true.

Truth tables for specific forms

Truth tables are constructed for forms like ~p ∧ q, p ∨ ~q, (p ∧ q) ∨ r, and p ∧ (q ∨ r) to show all valuation outcomes.