Propositions and Truth Tables

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

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

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”) Table 1. The Truth Table for the Negation of a Proposition p  p T F F T p = The sky is blue.  p = It is not the case that the sky is blue.  p = The sky is not blue.

Conjunction of p and q: 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. 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

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. 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

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

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

Implications 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

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

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

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” 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

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.

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)

(T/F) Conditional (T/F) Converse (T/F)Inverse (T/F)Contrapositive p m  C = 100° q  ABC is obtuse 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°

(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 If  ABC is equilateral, then it is isosceles p  ABC is equilateral q it is isosceles

(T/F) Conditional (T/F) Converse (T/F)Inverse (T/F)Contrapositive p G is the midpoint of KL q GQ bisects KL 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.

Let R = “I work at this school” and D = “My name is Ms. D” Translate the following symbols into sentences, and indicate T/F: a.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.

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.”

FTTFFTTF FTTFFTTF TTFTTTFT TFFTTFFT FTFTFTFT TFTFTFTF FFTTFFTT FTFTFTFT TFTTTFTT TFFTTFFT FFTFFFTF FFTTFFTT TFTFTFTF TFTFTFTF TFFTTFFT TTFTTTFT FTFTFTFT TFFTTFFT FTFTFTFT TFFTTFFT 6. (p ^ w) ~(p ^ w) 7. p (w  v) p  (w  v) 8. (v  t) r (v  t)  r 9. (k  p) g g  (k  p)10. ~w  r t (~w  r)  t 11. ~b (k  p) (k  p)  ~b 12. (n  j) e (n  j)  e 13. u (~h  c) (~h  c)  u 14 (w  ~ ~r) ~(w  ~ ~ r) 15. f  r ~t ( f  r)  ~t 16. (j  u) ~y ~y  (j  u) (j  u)  ~y TTFTTTFT TTFTTTFT TFTTTFTT TFTTTFTT FTTTFTTT FFTFFFTF TFFTTFFT TTTFTTTF FFFTFFFT TTFTTTFT FFTTFFTT FFFFFFFF