Propositions and Truth Tables

1. Propositions and Truth Tables

2. Proposition: Makes a claim that may be either true or false; it must have the structure of a complete sentence.

3. Are these propositions?Over the mountain and through the woods. All apples are fruit. The quick, brown fox. Are you here? 2 + 3 = 23 NO YES NO NO YES

4. Table 1. The Truth Table for the Negation of a Proposition p p T F F T Negation of p Let p be a proposition. The statement “It is not the case that p” is also a proposition, called the “negation of p” or p (read “not p”) p = The sky is blue. p = It is not the case that the sky is blue. p = The sky is not blue.

5. Table 2. The Truth Table for the Conjunction of two propositions p q pq T T T T F F F T F F F F Conjunction of p andq: AND Let p and q be propositions. The proposition “p and q,” denoted by pq is true when both p and q are true and is false otherwise. This is called the conjunction of p and q.

6. Table 3. The Truth Table for the Disjunction of two propositions p q pq T T T T F T F T T F F F Disjunction of p and q: OR Let p and q be propositions. The proposition “p or q,” denoted by pq, is the proposition that is false when p and q are both false and true otherwise.

7. Two types of OR INCLUSIVE OR means “either or both” EXCLUSIVE OR means “one or the other, but not both”

8. INCLUSIVE OR means “either or both” EXCLUSIVE OR means “one or the other, but not both” p q pq p q pq T T T T F T F T T F F F T T F T F T F T T F F F Two types of Disjunction of p and q: OR

9. If p, then q p implies q if p, q p only if q p is sufficient for q q if p q whenever p q is necessary for p Proposition p = antecedent Proposition q = consequent Implications

10. Conditional p  q Contrapositive of p  q is the proposition q  p. Converse, Inverse, Contrapositive p You are sleeping q you are breathing If you are sleeping, then you are breathing. If you are breathing, then you are sleeping. Converse of p  q is q  p Inverse of p  q is p  q If you are not sleeping, then you are not breathing. If you are not breathing, then you are not sleeping

11. Conditional p  q Contrapositive of p  q is the proposition q  p Find the conditional, converse, inverse and contrapositive: p The sun is shinning q it is warm outside If the sun is shining, then it is warm outside. Converse of p  q is q  p If it is warm outside, then the sun is shining. Inverse of p  q is p  q If the sun is not shining, then it is not warm outside. If it is not warm outside, the sun is not shining.

12. Table 6. The Truth Table for the biconditional pq. p q pq T T T T F F F T F F F T Biconditional Let p and q be propositions. The biconditional pq is the proposition that is true when p and q have the same truth values and is false otherwise. “p if and only if q, p is necessary and sufficient for q”

13. Logical Equivalence • An important technique in proofs is to replace a statement with another statement that is “logically equivalent.” • Tautology: compound proposition that is always true regardless of the truth values of the propositions in it. • Contradiction: Compound proposition that is always false regardless of the truth values of the propositions in it.

14. Logically Equivalent • Compound propositions P and Q are logically equivalent if PQ is a tautology. In other words, P and Q have the same truth values for all combinations of truth values of simple propositions. • This is denoted: PQ (or by P Q)

15. p mC = 100° q ABC is obtuse (T/F) Conditional (T/F) Converse (T/F) Inverse (T/F) Contrapositive If mC = 100°, then ABC is obtuse. If ABC is obtuse then mC = 100°, If mC ≠ 100°, then ABC is not obtuse. If ABC is not obtuse, then mC ≠100°

16. p ABC is equilateral q it is isosceles If ABC is equilateral, then it is isosceles (T/F) Conditional (T/F) Converse (T/F) Inverse (T/F) Contrapositive If ABC is isosceles, then it is equilateral If ABC is not equilateral, then it is not isosceles If ABC is not isosceles, then it is not equilateral

17. If G is the midpoint of KL, then GQ bisects KL. If GQ bisects KL, G is the midpoint of KL. If G is not the midpoint of KL, then GQ does not bisects KL. If GQ does not bisects KL then G is not the midpoint of KL. p G is the midpoint of KL q GQ bisects KL (T/F) Conditional (T/F) Converse (T/F) Inverse (T/F) Contrapositive

18. Let R = “I work at this school” • and D = “My name is Ms. D” • Translate the following symbols into sentences, and indicate T/F: • D  R: ___ (T/F) • b. R  D: ___ (T/F) • c. ~D  ~R: ___ (T/F) If my name is Ms. D, then I work at this school. If I work at this school, then my name is Ms. D. If my name is not Ms. D, then I do not work at this school.

19. 5. Given: a. If today is warm, the pool will be crowded. b. If it rains today, the pool will not be crowded. c. Either today is warm or I will wear a long-sleeved shirt. d. It will not rain today. Using W, P, R, S, & proper connectives (~, , etc.), express each sentence into symbolic form. Let W represent “Today is warm.” Let P represent “The pool will be crowded.” Let R represent “It rains today.” Let S represent “I will wear a long-sleeved shirt.”

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