Propositional Logic ITCS 2175 (Rosen Section 1.1, 1.2)

Proposition A proposition is a statement that is either true or false, but not both. Atlanta was the site of the 1996 Summer Olympic games. 1+1 = 2 3+1 = 5 What will my ITCS2175 grade be?

Definition 1. 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.

Definition 2. Conjunction of p and q 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

Definition 3. Disjunction of p and q 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

Definition 4. Exclusive or of p and q Let p and q be propositions. The exclusive or of p and q, denoted by pq, is the proposition that is true when exactly one of p and q is true and is false otherwise. Table 4. The Truth Table for the Exclusive OR of two propositions p q pq T T F T F T F T T F F F

Definition 5. Implication pq Let p and q be propositions. The implication pq is the proposition that is false when p is true and q is false, and true otherwise. In this implication p is called the hypothesis (or antecedent or premise) and q is called the conclusion (or consequence). Table 5. The Truth Table for the Implication of pq. p q pq T T T T F F F T T F F T

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 Not the same as the if-then construct used in programming languages such as If p then S

Implications How can both p and q be false, and pq be true? Think of p as a “contract” and q as its “obligation” that is only carried out if the contract is valid. Example: “If you make more than $25,000, then you must file a tax return.” This says nothing about someone who makes less than $25,000. So the implication is true no matter what someone making less than $25,000 does. Another example: p: Bill Gates is poor. q: Pigs can fly. pq is always true because Bill Gates is not poor. Another way of saying the implication is “Pigs can fly whenever Bill Gates is poor” which is true since neither p nor q is true.

Related Implications Converse of p  q is q  p Contrapositive of p  q is the proposition q  p

Definition 6. 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

Practice p: You learn the simple things well. q: The difficult things become easy. The difficult things become easy but you did not learn the simple things well. You learn the simple things well but the difficult things did not become easy. You do not learn the simple things well. If you learn the simple things well then the difficult things become easy. If you do not learn the simple things well, then the difficult things will not become easy.

Practice p: You learn the simple things well. q: The difficult things become easy. The difficult things become easy but you did not learn the simple things well. You learn the simple things well but the difficult things did not become easy. You do not learn the simple things well. If you learn the simple things well then the difficult things become easy. If you do not learn the simple things well, then the difficult things will not become easy. p q  p pq p  q p  q

Truth Table Puzzle Steve would like to determine the relative salaries of three coworkers using two facts (all salaries are distinct): If Fred is not the highest paid of the three, then Janice is. If Janice is not the lowest paid, then Maggie is paid the most. Who is paid the most and who is paid the least?

p : Janice is paid the most. q: Maggie is paid the most. r: Fred is paid the most. s: Janice is paid the least. p q r s rp s q (rp) (sq) T F F F T F F F T F T F T F F F T T T T T F T F F F T F F F T F T F F Fred, Maggie, Janice If Fred is not the highest paid of the three, then Janice is. If Janice is not the lowest paid, then Maggie is paid the most. Only 5 entries because the other combinations are contradictory (i.e., Janice and Fred cannot be paid the most at the same time if salaries are distinct.

p : Janice is paid the most. q: Maggie is paid the most. r: Fred is paid the most. s: Janice is paid the least. p q r s rp s q (rp) (sq) T F F F T T T F T F T F T F F F T T T F F F T F F F T F F F T F T T T Fred, Janice, Maggie or Janice, Maggie, Fred or Janice, Fred, Maggie If Fred is not the highest paid of the three, then Janice is. If Janice is the lowest paid, then Maggie is paid the most.

Bit Operations A computer bit has two possible values: 0 (false) and 1 (true). A variable is called a Boolean variable if its value is either true or false. Bit operations correspond to the logical connectives:  OR  AND  XOR Information can be represented by bit strings, which are sequences of zeros and ones, and manipulated by operations on the bit strings.

Truth tables for the bit operations OR, AND, and XOR  0 1 0 0 1 1 1 0  0 1 0 0 1 1 1 1  0 1 0 0 0 1 0 1

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)

Example: DeMorgans Prove that (pq)  (p  q) p q (pq) (pq) p q (p  q) T T T F F T F F T F F F F T F F T F T F T F F F T T T T

Illustration of De Morgan’s Law (pq) p q

Illustration of De Morgan’s Law p p

Illustration of De Morgan’s Law q q

Illustration of De Morgan’s Law p  q p q

Example: Distribution Prove that: p  (q  r)  (p  q)  (p  r) p q r qr p(qr) pq pr (pq)(pr) T T T T T T T T T T F F T T T T T F T F T T T T T F F F T T T T F T T T T T T T F T F F F T F F F F T F F F T F F F F F F F F F

Prove: pq(pq)  (qp) p q pq pq qp (pq)(qp) T T T T T T T F F F T F F T F T F F F F T T T T We call this biconditional equivalence.

List of Logical Equivalences pT  p; pF  p Identity Laws pT  T; pF  F Domination Laws pp  p; pp  p Idempotent Laws (p)  p Double Negation Law pq  qp; pq  qp Commutative Laws (pq) r  p (qr); (pq)  r  p  (qr) Associative Laws

List of Equivalences p(qr)  (pq)(pr) Distribution Laws (pq)(p  q) De Morgan’s Laws (pq)(p  q) Miscellaneous p  p  T Or Tautology p  p  F And Contradiction (pq)  (p  q) Implication Equivalence pq(pq)  (qp) Biconditional Equivalence