Propositional Logic

Propositional Logic

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 CS1050 grade be?

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

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 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 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 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 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 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 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 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).

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 Inverse of p  q Is the proposition p  q

Example • implication: “If it rains today, I will go to college tomorrow” • Converse: I will go to college tomorrow only if it rains today • Contrapositive : If I do not go to college tomorrow, then it will not have rained today • Inverse : If it does not rain today, then I will not go to college tomorrow

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 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”

Practice p: You learn the simple things well. q: The difficult things 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. • 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. 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.

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.

Well formed Formula (WFF) • A well formed formula can be produced using following rules: • Rule 1 : A statement variable itself is a WFF • Rule 2 : If p is WFF, then p is WFF • Rule 3 : If p and q are WFF then (pq), (p  q), (p  q) and (pq) are also WFF • Rule 4 : A string of symbols consisting of statement variables, connectives and parentheses is said to be WFF iff it can be produced by applying rule 1, 2 and 3 finitely many times

Bit Operations A computer bit has two possible values: 0 (false) and 1 (true). A variable is called a Boolean variable is 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.

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

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. Eg. p  p • Contradiction: Compound proposition that is always false regardless of the truth values of the propositions in it. Eg. p p

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r)  (pq)(pr) (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

Propositional Logic