Definition 2. Conjunction of p and q

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1 Proposition Propositional Logic CPSC 2070 Discrete Structures Rosen (6 th Ed.) 1.1, 1.2 A proposition is a statement that is either true or false, but not both. Clemson will defeat Georgia in football this fall. 1+1 = = 5 What will be my grade in CPSC 2070? LOGICAL We can define operations on propositions! Definition 1. Negation of p Let p be a proposition. he statement It is not the case that p is also a proposition, called the negation of p or p (read not p ) p = he sky is blue. p = It is not the case that the sky is blue. p = he sky is not blue. able 1. he ruth able for the Negation of a Proposition p p Definition 2. Conjunction of p and q propositions. he proposition p and q, denoted by p q is true when both p and q are true and is false otherwise. his is called the conjunction of p and q. able 2. he ruth able for the Conjunction of two propositions p q p q Definition 3. Disjunction of p and q able 3. he ruth able for the Disjunction of two propositions p q p q propositions. he proposition p or q, denoted by p q, is the proposition that is false when p and q are both false and true otherwise. 1

2 Definition 4. Exclusive or of p and q able 4. he ruth able for the Exclusive OR of two propositions p q p q propositions. he 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. Definition 5. Implication p q propositions. he 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). Politician Example able 5. he ruth able for the Implication of p q. p q p q Implications Related Implications If p, then q p implies q if p,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 Converse of p q is the proposition q p Inverse of p q is the proposition p q Contrapositive of p q is the proposition q p Definition 6. Biconditional able 6. he ruth able for the biconditional p q. p q p q propositions. he 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 Compound Propositions We can also combine operations to create compound propositions such as: (p q) q est all possible combinations of / Don t try to do it all in your head! p q (p q) (p q) (p q) q 2

3 Special ypes of Compound Propositions Contradiction: Compound proposition that is always false regardless of the truth values of the propositions in it. p p is a contradiction autology: compound proposition that is always true regardless of the truth values of the propositions in it. p p is a tautology Practice with English Sentences p: You learn the simple things well. q: he 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. he 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. Practice with English Sentences p: You learn the simple things well. q: he difficult things become easy. You do not learn the simple things well. p If you learn the simple things well then the difficult things become easy. p q If you do not learn the simple things well, then the difficult things will not become easy. p q he difficult things become easy but you did not learn the simple things well. q p You learn the simple things well but the difficult things did not become easy. p q Some Applications of Propositional Logic ruth able Puzzle Steve would like to determine the relative salaries of three coworkers using two facts: If red 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: red is paid the most. s: Janice is paid the least. p q r s r p s q ( r p) ( s q) red, Maggie, Janice If Janice is not the lowest If red is not the paid, then Maggie is paid highest paid of the the most. three, then Janice is. 3

4 Knights and Knaves Puzzles On a remote island there live Knights and Knaves. Knights always tell the truth and Knaves always lie. You meet two people on the island A and B. A says: B and I are both Knights. B says: A is a Knave. Determine, if possible, which group A and B belong to. Let a be the statement that A is a Knight and b be the statement that B is a Knight. A says B and I are both Knights. If A is a Knight then a is true and the statement (a b) is true. If A is a Knave, then a is false and the statement is false. hese are the two cases that are true in that they are possible solutions to the puzzle. We can code this with the biconditional operator. (Recall that p q is true when both propositions are true or both propositions are false.) B says A is a Knave. Using the same logic add b a to the table. a (A is a Knight) b (B is a Knight) (a b) a (a b) b a A says At least one of us is a Knave and B says nothing. a (A is a Knight) b (B is a Knight) a b a ( a b) he Hat Puzzle hree students who made an A in CPSC 2070 are told to stand in a straight line, one in front of the other. A hat is put on each of their heads. hey are told that each of these hats was selected from a group of five hats: two black hats and three white hats. he first student, standing at the front of the line, can t see either of the students behind her or their hats. he second student, in the middle, can see only the first student and her hat. he last student, at the rear, can see both other students and their hats. None of them can see the hat on their own head. hey are asked to deduce its color. he last student in line is asked if he knows the color of his hat and says he cannot be sure. he second student in line is asked if he knows the color of his hat and says he cannot be sure. he student at the front of the line then says: My hat is white. She is correct. How did she come to this conclusion? Bit Operations Last Second irst W W W W W B W B W W B B B W W B W B B B W 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 zeroes and ones, and manipulated by operations on the bit strings. 4

5 ruth tables for the bit operations OR, AND, and XOR Binary Math Review Decimal has digits Binary has digits = = = = = 100 Program to add 3 binary digits p q r Output Output 1 = (p q) (p r) (q r) Output 2 Output 2 = p q r 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. his is denoted: P Q ull Adder: Computers contain switches (logic gates) corresponding to our logic operations Example: DeMorgans Prove that (p q) ( p q) p q (p q) (p q) 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) 5

6 Prove: p q (p q) (q p) p q p q p q q p (p q) (q p) We call this biconditional equivalence. List of Logical Equivalences able 6 in Section 1.2 p p; p p Identity Laws p ; p 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) Distributive Laws p (q r) (p q) (p r) (p q) ( p q) (p q) ( p q) De Morgan s Laws p (p q) p Absorption Laws p (p q) p p p p p (p q) ( p q) Negation Laws Or autology; And Contradiction Implication Equivalence 6