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Conjunctions

3.2 – Truth Tables and Equivalent Statements. Conjunctions. Truth Values. The truth values of component statements are used to find the truth values of compound statements.

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Conjunctions

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  1. 3.2 – Truth Tables and Equivalent Statements Conjunctions Truth Values The truth values of component statements are used to find the truth values of compound statements. 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.

  2. 3.2 – Truth Tables and Equivalent Statements Conjunction Truth Table p and q

  3. 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 12 < 9 q is false The truth value for p ˄ q is false

  4. 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

  5. 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 12 < 9 q is false The truth value for p ˅ q is true

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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 The truth value for the statement is false.

  15. 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)

  16. 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)

  17. 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)

  18. 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)

  19. 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) The truth value for the statement is true.

  20. 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)

  21. 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)

  22. 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) Yes

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