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Questions and Answers
What is the purpose of truth tables in propositional logic?
What is the purpose of truth tables in propositional logic?
What is the notation for the negation operator in propositional logic?
What is the notation for the negation operator in propositional logic?
What is the result of the implication operator p → q if p is false and q is true?
What is the result of the implication operator p → q if p is false and q is true?
Which law states that p ∧ q ≡ q ∧ p?
Which law states that p ∧ q ≡ q ∧ p?
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What is the result of the equivalence operator p ⇔ q if p is true and q is false?
What is the result of the equivalence operator p ⇔ q if p is true and q is false?
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What is the result of applying De Morgan's Law to ¬(p ∧ q)?
What is the result of applying De Morgan's Law to ¬(p ∧ q)?
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What is the term for the statement P in an implication P → Q?
What is the term for the statement P in an implication P → Q?
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Which rule states that if P → Q and P, then Q?
Which rule states that if P → Q and P, then Q?
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What is the equivalent statement to P → Q?
What is the equivalent statement to P → Q?
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What can be concluded if P → Q and Q → R?
What can be concluded if P → Q and Q → R?
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What is the term for the statement Q in an implication P → Q?
What is the term for the statement Q in an implication P → Q?
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Which rule states that if P → Q and not Q, then not P?
Which rule states that if P → Q and not Q, then not P?
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Study Notes
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)
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Test your understanding of propositional logic, including propositions, logical operators, and truth tables.
