3.2 – Truth Tables and Equivalent Statements

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Presentation transcript:

3.2 – Truth Tables and Equivalent Statements Truth Values The truth values of component statements are used to find the truth values of compound statements. Conjunctions The truth values of the conjunction p and q (p ˄ q), are given in the truth table on the next slide. The connective “and” implies “both.” Truth Table A truth table shows all four possible combinations of truth values for component statements.

Conjunction Truth Table 3.2 – Truth Tables and Equivalent Statements Conjunction Truth Table p and q p q p ˄ q T T T T F F F T F F

Finding the Truth Value of a Conjunction 3.2 – Truth Tables and Equivalent Statements Finding the Truth Value of a Conjunction If p represent the statement 4 > 1 and q represent the statement 12 < 9, find the truth value of p ˄ q. p and q 4 > 1 p is true p q p ˄ q T T T T F F F T F F 12 < 9 q is false The truth value for p ˄ q is false

3.2 – Truth Tables and Equivalent Statements Disjunctions The truth values of the disjunction p or q (p ˅ q) are given in the truth table below. The connective “or” implies “either.” Disjunction Truth Table p or q p q p ˅ q T T T T F F T F F F

Finding the Truth Value of a Disjunction 3.2 – Truth Tables and Equivalent Statements Finding the Truth Value of a Disjunction If p represent the statement 4 > 1, and q represent the statement 12 < 9, find the truth value of p ˅ q. p or q 4 > 1 p is true p q p ˅ q T T T T F F T F F F 12 < 9 q is false The truth value for p ˅ q is true

3.2 – Truth Tables and Equivalent Statements Negation The truth values of the negation of p ( ̴ p) are given in the truth table below. not p p ̴ p T F

Example: Constructing a Truth Table 3.2 – Truth Tables and Equivalent Statements Example: Constructing a Truth Table Construct the truth table for: p ˄ (~ p ˅ ~ q) A logical statement having n component statements will have 2n rows in its truth table. 22 = 4 rows p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q) T T T F F T F F

Example: Constructing a Truth Table 3.2 – Truth Tables and Equivalent Statements Example: Constructing a Truth Table Construct the truth table for: p ˄ (~ p ˅ ~ q) A logical statement having n component statements will have 2n rows in its truth table. 22 = 4 rows p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q) T T F T F F T T F F

Example: Constructing a Truth Table 3.2 – Truth Tables and Equivalent Statements Example: Constructing a Truth Table Construct the truth table for: p ˄ (~ p ˅ ~ q) A logical statement having n component statements will have 2n rows in its truth table. 22 = 4 rows p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q) T T F T F T F T F F

Example: Constructing a Truth Table 3.2 – Truth Tables and Equivalent Statements Example: Constructing a Truth Table Construct the truth table for: p ˄ (~ p ˅ ~ q) A logical statement having n component statements will have 2n rows in its truth table. 22 = 4 rows p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q) T T F T F T F T F F

Example: Constructing a Truth Table 3.2 – Truth Tables and Equivalent Statements Example: Constructing a Truth Table Construct the truth table for: p ˄ (~ p ˅ ~ q) A logical statement having n component statements will have 2n rows in its truth table. 22 = 4 rows p q ~ p ~ q ~ p ˅ ~ q p ˄ (~ p ˅ ~ q) T T F T F T F T F F

Example: Mathematical Statements 3.2 – Truth Tables and Equivalent Statements Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. ̴ p ˄ ̴ q p q ̴ p ̴ q T T T F F T F F ̴ p ˄ ̴ q

Example: Mathematical Statements 3.2 – Truth Tables and Equivalent Statements Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. ̴ p ˄ ̴ q p q ̴ p ̴ q T T F F T F F T ̴ p ˄ ̴ q

Example: Mathematical Statements 3.2 – Truth Tables and Equivalent Statements Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. ̴ p ˄ ̴ q p q ̴ p ̴ q T T F F T F F T ̴ p ˄ ̴ q F T The truth value for the statement is false.

Example: Mathematical Statements 3.2 – Truth Tables and Equivalent Statements Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. ( ̴ p ˄ r) ˅ ( ̴ q ˄ p) p q r ̴ p ̴ q ̴ r T T T T T F T F T T F F F T T F T F F F T F F F ̴ p ˄ r ̴ q ˄ p ˅

Example: Mathematical Statements 3.2 – Truth Tables and Equivalent Statements Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. ( ̴ p ˄ r) ˅ ( ̴ q ˄ p) p q r ̴ p ̴ q ̴ r T T T F F F T T F F F T T F T F T F T F F F T T F T T T F F F T F T F T F F T T T F F F F T T T ̴ p ˄ r ̴ q ˄ p ˅

Example: Mathematical Statements 3.2 – Truth Tables and Equivalent Statements Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. ( ̴ p ˄ r) ˅ ( ̴ q ˄ p) p q r ̴ p ̴ q ̴ r T T T F F F T T F F F T T F T F T F T F F F T T F T T T F F F T F T F T F F T T T F F F F T T T ̴ p ˄ r ̴ q ˄ p F T ˅

Example: Mathematical Statements 3.2 – Truth Tables and Equivalent Statements Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. ( ̴ p ˄ r) ˅ ( ̴ q ˄ p) p q r ̴ p ̴ q ̴ r T T T F F F T T F F F T T F T F T F T F F F T T F T T T F F F T F T F T F F T T T F F F F T T T ̴ p ˄ r ̴ q ˄ p F T ˅

Example: Mathematical Statements 3.2 – Truth Tables and Equivalent Statements Example: Mathematical Statements If p represent the statement 4 > 1, and q represent the statement 12 < 9, and r represent 0 < 1, decide whether the statement is true or false. ( ̴ p ˄ r) ˅ ( ̴ q ˄ p) p q r ̴ p ̴ q ̴ r T T T F F F T T F F F T T F T F T F T F F F T T F T T T F F F T F T F T F F T T T F F F F T T T ̴ p ˄ r ̴ q ˄ p F T ˅ F T The truth value for the statement is true.

Equivalent Statements 3.2 – Truth Tables and Equivalent Statements Equivalent Statements Two statements are equivalent if they have the same truth value in every possible situation. Are the following statements equivalent? ~ p ˄ ~ q and ̴ (p ˅ q) p q ~ p ˄ ~ q ̴ (p ˅ q) T T T F F T F F

Equivalent Statements 3.2 – Truth Tables and Equivalent Statements Equivalent Statements Two statements are equivalent if they have the same truth value in every possible situation. Are the following statements equivalent? ~ p ˄ ~ q and ̴ (p ˅ q) p q ~ p ˄ ~ q ̴ (p ˅ q) T T F T F F T F F T

Equivalent Statements 3.2 – Truth Tables and Equivalent Statements Equivalent Statements Two statements are equivalent if they have the same truth value in every possible situation. Are the following statements equivalent? ~ p ˄ ~ q and ̴ (p ˅ q) p q ~ p ˄ ~ q ̴ (p ˅ q) T T F T F F T F F T Yes