Propositional Logic Basics Quiz

Propositions

• A proposition is a declarative statement that is either true or false.

Negation

• A negation is a logical connective that flips the truth value of a proposition.
• For example, the negation of "The sky is blue" is "The sky is not blue".

Truth Tables

• Truth tables are primarily used to evaluate the truth values of complex propositions based on the truth values of their components.
• They help determine the truth value of a compound proposition for all possible combinations of truth values of its individual components.

Conjunction

• A proposition formed by combining two simpler propositions using the word 'and' is called a conjunction.
• The conjunction of propositions p and q is denoted by p ∧ q.

Conjunction of True and False

• The result of the conjunction of two propositions T and F is F.

Negation Symbol

• The symbol that represents negation in propositional logic is ¬.

Disjunction

• The disjunction of propositions p and q is denoted by p ∨ q.

Logical Connectives

• Conjunction (∧): Represents "and"
• p ∧ q is true only if both p and q are true.
• Disjunction (∨): Represents "or"
• p ∨ q is true if at least one of p or q is true.
• Negation (¬): Represents "not"
• ¬p is true if p is false.
• Conditional (→): Represents "if...then"
• p → q is true unless p is true and q is false.

Conditional Statement

• The connective that describes the statement "If it is raining, then I will stay indoors" is the conditional connective (→).

Tautology

• The proposition that is always true is denoted by T (or sometimes 1).

Disjunction of False Propositions

• The truth value of the expression p ∨ q when both p and q are false is F.

Universal Quantifier

• A universal quantifier asserts that a property holds for all elements of a given domain.
• It is symbolized by ∀.

Predicate Logic

• Predicate Logic can represent all logical statements that can be formed with propositional logic.
• This is because predicate logic has the power to handle quantifiers and relations, which can be used to express more complex statements.

Predicate

• In the expression P(x) if P is defined as 'x is greater than zero', the predicate is "x is greater than zero".

Instantiation

• The process of replacing variables in a propositional function with values from their domain is called instantiation.

Propositional Function Truth Values

• P(x): x is greater than 5
• P(3): False
• P(7): True
• Q(x): x is an even number
• Q(4): True
• Q(9): False

Valid Argument Definition

• A valid argument in logic is one where the conclusion logically follows from the premises.
• This means that if all the premises are true, then the conclusion must also be true.

Existential Quantifier Negation

• Existential quantifiers can be negated by rephrasing them as universal quantifiers.

Existential Quantifier Example

• "There exists a person who is over 100 years old"
• This statement can be represented by the existential quantifier: ∃x (Person(x) ∧ Age(x) > 100).