2.1 Logical Equivalence and Truth Tables 2.1 Logical Equivalence and Truth Tables 1 / 9 The truth table for a given statement form displays the truth values that correspond to all possible combinations of truth values for its component statement variables. Statement (propositional) forms Definition A statement form (or propositional form) is an expression made up of statement variables (such as p,q, and r) and logical connectives (such as ∼; ^, and _) that becomes a statement when actual statements are substituted for the component statement variables. 2.1 Logical Equivalence and Truth Tables 2 / 9 Statement (propositional) forms Definition A statement form (or propositional form) is an expression made up of statement variables (such as p,q, and r) and logical connectives (such as ∼; ^, and _) that becomes a statement when actual statements are substituted for the component statement variables. The truth table for a given statement form displays the truth values that correspond to all possible combinations of truth values for its component statement variables. 2.1 Logical Equivalence and Truth Tables 2 / 9 2 p _ (q ^ r) 3 (p _ q) ^ (p _ r) Truth Tables for statement forms Examples Find the truth tables for the following statement forms: 1 p_ ∼ q 2.1 Logical Equivalence and Truth Tables 3 / 9 3 (p _ q) ^ (p _ r) Truth Tables for statement forms Examples Find the truth tables for the following statement forms: 1 p_ ∼ q 2 p _ (q ^ r) 2.1 Logical Equivalence and Truth Tables 3 / 9 Truth Tables for statement forms Examples Find the truth tables for the following statement forms: 1 p_ ∼ q 2 p _ (q ^ r) 3 (p _ q) ^ (p _ r) 2.1 Logical Equivalence and Truth Tables 3 / 9 The logical equivalence of statement forms P and Q is denoted by writing P ≡ Q. Two statements are called logically equivalent if, and only if, they have logically equivalent forms when identical component statement variables are used to replace identical component statements. Logical Equivalence Definition Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution for their statement variables. 2.1 Logical Equivalence and Truth Tables 4 / 9 Two statements are called logically equivalent if, and only if, they have logically equivalent forms when identical component statement variables are used to replace identical component statements. Logical Equivalence Definition Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution for their statement variables. The logical equivalence of statement forms P and Q is denoted by writing P ≡ Q. 2.1 Logical Equivalence and Truth Tables 4 / 9 Logical Equivalence Definition Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution for their statement variables. The logical equivalence of statement forms P and Q is denoted by writing P ≡ Q. Two statements are called logically equivalent if, and only if, they have logically equivalent forms when identical component statement variables are used to replace identical component statements. 2.1 Logical Equivalence and Truth Tables 4 / 9 2 Show that ∼ (p _ q) ≡∼ p^ ∼ q. 3 Show that ∼ (p ^ q) and ∼ p^ ∼ q are not logically equivalent. Examples Examples (de Morgan's Laws) 1 We have seen that ∼ (p ^ q) and ∼ p_ ∼ q are logically equivalent. 2.1 Logical Equivalence and Truth Tables 5 / 9 3 Show that ∼ (p ^ q) and ∼ p^ ∼ q are not logically equivalent. Examples Examples (de Morgan's Laws) 1 We have seen that ∼ (p ^ q) and ∼ p_ ∼ q are logically equivalent. 2 Show that ∼ (p _ q) ≡∼ p^ ∼ q. 2.1 Logical Equivalence and Truth Tables 5 / 9 Examples Examples (de Morgan's Laws) 1 We have seen that ∼ (p ^ q) and ∼ p_ ∼ q are logically equivalent. 2 Show that ∼ (p _ q) ≡∼ p^ ∼ q. 3 Show that ∼ (p ^ q) and ∼ p^ ∼ q are not logically equivalent. 2.1 Logical Equivalence and Truth Tables 5 / 9 Example: p_ ∼ p. Example: p^ ∼ p. Tautologies and Contradictions Definition A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables. A contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. 2.1 Logical Equivalence and Truth Tables 6 / 9 Example: p^ ∼ p. Tautologies and Contradictions Definition A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables. Example: p_ ∼ p. A contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. 2.1 Logical Equivalence and Truth Tables 6 / 9 Tautologies and Contradictions Definition A tautology is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables. Example: p_ ∼ p. A contradiction is a statement form that is always false regardless of the truth values of the individual statements substituted for its statement variables. Example: p^ ∼ p. 2.1 Logical Equivalence and Truth Tables 6 / 9 Logical Equivalence 2.1 Logical Equivalence and Truth Tables 7 / 9 Logical Equivalence 2.1 Logical Equivalence and Truth Tables 8 / 9 Examples Examples Use the logical equivalence of Theorem 2.1.1 to prove that the following are logical equivalences: 1 (∼ p _ q) ^ (∼ q) ≡∼ (p _ q). 2 (p^ ∼ q) _ (∼ p _ q) ≡ t. 2.1 Logical Equivalence and Truth Tables 9 / 9.