Truth Tables, Exams of Logic

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Truth Tables

I. Negation (~ : not) expresses the opposite truth value. p ~ p T F F T II. Conjunction (Ʌ : and) is true only III. Disjunction ( V : or) is false only when both statements are true. when both statements are false. p q p^ Ʌ^ p T T T T F F F T F F F F p q p^ V^ q T T T T F T F T T F F F IV. Conditional (→ : if-then) is false only V. Biconditional (↔ : if and only if) is when the antecedent (1st) is true and true only when the component the component (2nd) is false. statements have the same truth value. p q p → q T T T T F F F T T F F T p q p ↔ q T T T T F F F T F F F T Fall 2017

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VI. Order to perform logic operators in truth tables:

• Parenthesis o ( )
• Negation o ~
• Conjunction o Ʌ
• Disjunction o V
• Conditional and Biconditional o → and ↔ **VII. Truth Table Examples:

1. ~** p Ʌ p 2. ~( p Ʌ q ) p ~ p ~ p Ʌ p T F F F T F p q (p Ʌ q) ~ ( p Ʌ q ) T T T F T F F T F T F T F F F T 3. p V ~q 4. ~(p V q) p q ~q p V ~q p q (p V q) ~(p V q) T T F T T T T F T F T T T F T F F T F F F T T F

F F F F T

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7. Determine the truth value for ~p Ʌ ( ~q V r) when p is false, q is true, and r is false. ~p Ʌ (~q V r) Original statement ~F Ʌ (~T V F) Original statement with truth values ~F Ʌ (F V F) Perform the negation in the parenthesis ~F Ʌ (F) Finish the parenthesis T Ʌ (F) Perform the negation False Perform the conjunction 8. Determine the truth value for ~p V ~ ( q V r) when p is false, q is true, and r is false. ~p V ~(q V r) Original statement ~F V ~(T V F) Original statement with truth values ~F V ~(T) Perform the parenthesis T V F Perform the negations True Perform the disjunction 9. Determine the truth value for ( ~p Ʌ q) ↔ ~r when p is false, q is true, and r is false. (~p Ʌ q) ↔ ~r Original statement (~F Ʌ T) ↔ ~F (^) Original statement with truth values (T Ʌ T) ↔ ~F Perform the negation in the parenthesis (T) ↔ ~F Finish the parenthesis

T ↔ T Perform the negation True Perform the biconditional Fall 2017 M-T