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2.1 and Tables

2.1 Logical Equivalence and Truth Tables 1 / 9 The for a given 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 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 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

Definition A is a statement form that is always true regardless of the truth values of the individual statements substituted for its statement variables.

A is a statement form that is always 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 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