Propositional Logic Quiz

Propositional Logic

Introduction

• Propositional Logic is a branch of logic that deals with statements that can be either true or false.
• It involves the study of logical operators and their interactions.

Basic Concepts

• Propositions: Statements that can be either true (T) or false (F).
• Logical Operators: Symbols used to connect propositions to form new statements.
• Truth Tables: Tables used to visualize the truth values of propositions.

Logical Operators

Negation (NOT)

• Denoted by ¬
• Reverses the truth value of a proposition
• Example: ¬p (not p)

Conjunction (AND)

• Denoted by ∧
• Returns true if both propositions are true
• Example: p ∧ q (p and q)

Disjunction (OR)

• Denoted by ∨
• Returns true if at least one proposition is true
• Example: p ∨ q (p or q)

Implication (IF-THEN)

• Denoted by →
• Returns true if the antecedent is false or the consequent is true
• Example: p → q (if p then q)

Equivalence (IF-AND-ONLY-IF)

• Denoted by ⇔
• Returns true if both propositions have the same truth value
• Example: p ⇔ q (p if and only if q)

Laws of Propositional Logic

Commutative Laws

• p ∧ q ≡ q ∧ p
• p ∨ q ≡ q ∨ p

Associative Laws

• (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
• (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)

Distributive Laws

• p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
• p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

De Morgan's Laws

• ¬(p ∧ q) ≡ ¬p ∨ ¬q
• ¬(p ∨ q) ≡ ¬p ∧ ¬q

Propositional Logic

• Propositional Logic is a branch of logic that deals with statements that can be either true or false.
• It involves the study of logical operators and their interactions.

Basic Concepts

• Propositions are statements that can be either true (T) or false (F).
• Logical Operators are symbols used to connect propositions to form new statements.
• Truth Tables are tables used to visualize the truth values of propositions.

Logical Operators

Negation (NOT)

• Denoted by ¬
• Reverses the truth value of a proposition
• Example: ¬p (not p)

Conjunction (AND)

• Denoted by ∧
• Returns true if both propositions are true
• Example: p ∧ q (p and q)

Disjunction (OR)

• Denoted by ∨
• Returns true if at least one proposition is true
• Example: p ∨ q (p or q)

Implication (IF-THEN)

• Denoted by →
• Returns true if the antecedent is false or the consequent is true
• Example: p → q (if p then q)

Equivalence (IF-AND-ONLY-IF)

• Denoted by ⇔
• Returns true if both propositions have the same truth value
• Example: p ⇔ q (p if and only if q)

Laws of Propositional Logic

Commutative Laws

• The order of propositions does not affect the result
• p ∧ q ≡ q ∧ p
• p ∨ q ≡ q ∨ p

Associative Laws

• The order in which operations are performed does not affect the result
• (p ∧ q) ∧ r ≡ p ∧ (q ∧ r)
• (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)

Distributive Laws

• The ∧ operator distributes over the ∨ operator
• p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
• The ∨ operator distributes over the ∧ operator
• p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)

De Morgan's Laws

• The ¬ operator can be distributed over ∧ and ∨ operators
• ¬(p ∧ q) ≡ ¬p ∨ ¬q
• ¬(p ∨ q) ≡ ¬p ∧ ¬q

Conditional Statements (If-Then Statements)

• A conditional statement has the form "If P, then Q" or "P → Q", read as "P implies Q".
• P is called the hypothesis or antecedent.
• Q is called the conclusion or consequent.

Properties of Conditional Statements

• Modus Ponens: If P → Q and P, then Q (if the hypothesis is true, the conclusion must also be true).
• Modus Tollens: If P → Q and not Q, then not P (if the conclusion is false, the hypothesis must also be false).
• Hypothetical Syllogism: If P → Q and Q → R, then P → R (if P implies Q and Q implies R, then P implies R).

Logical Equivalences of Conditional Statements

• Contrapositive: P → Q is equivalent to ¬Q → ¬P (not Q implies not P).
• Inverse: P → Q is equivalent to ¬P ∨ Q (not P or Q).
• Converse: P → Q does not imply Q → P (the converse is not necessarily true).

Examples of Conditional Statements

• If it rains, then the streets will be wet. (P: it rains, Q: the streets will be wet)
• If n is even, then n^2 is even. (P: n is even, Q: n^2 is even)

Test your understanding of propositional logic, including propositions, logical operators, and truth tables.