Introduction Propositional Logic. Discrete Mathematics Andrei Bulatov
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1 Introduction Propositional Logic Discrete Mathematics Andrei Bulatov
2 Discrete Mathematics Propositional Logic 2-2 What is Logic? Computer science is a mere continuation of logic by other means Georg Gottlob Contrariwise, continued Tweedledee, if it was so, it might be; and if it were so, it would be; but as it isn t, it ain t. That s logic Lewis Carroll, Through the Looking Glass Logic celebrates the unity of a pathological masculine self-identity that cannot listen and recognizes only negation but not difference Andrea Nye, Words of Power
3 Discrete Mathematics Propositional Logic 2-3 Use of Logic In mathematics and rhetoric: Give precise meaning to statements. Distinguish between valid and invalid arguments. Provide rules of `correct reasoning. Natural language can be very ambiguous `If you do your homework, then you ll get to watch the game. `If you don t do your homework, then you will not get to watch... `You do your homework, or you ll fail the exam. = `If you don t do your homework, then you ll fail the exam.
4 Discrete Mathematics Propositional Logic 2-4 Use of Logic (cntd) In computing: Derive new data / knowledge from existing facts Design of computer circuits. Construction of computer programs. Verification of correctness of programs and circuit design. Specification What the customer really needed How the Programmer understood it What the customer got
5 Discrete Mathematics Propositional Logic 2-5 Statements (Propositions) Propositional logic deals with statements and their truth values A statement is a declarative sentence that can be true or false Truth values are TRUTH (T or 1) and FALSE (F or 0). Examples: = 2 (statement, T) - The moon is made of cheese (statement, F) - Go home! (not statement, imperative) - What a beautiful garden! (not statement, exclamation) - y + 1 = 2 (not statement, uncertain) - The God exists (statement,?)
6 Discrete Mathematics Propositional Logic 2-6 Compound Statements Simplest statements are called primitive statements We shall use propositional variables to denote primitive statements, p, q, r, We cannot decide the truth value of a primitive statement. This is not what logic does. Instead we combine primitive statements by means of logic connectives into compound statements or formulas and look how the truth value of a compound statement depends on the truth values of the primitive statements it includes. We will denote compound statements by Φ, Ψ,
7 Discrete Mathematics Propositional Logic 2-7 Logic Connectives negation (not, ) `It is not true that at least one politician was honest conjunction (and, ) `In this room there is a lady, and in the other room there is a tiger disjunction (or, ) `Margaret Mitchell wrote `Gone with the Wind or I am going home implication (if, then, ) `If there is a tiger in this room, then there is no lady there exclusive or (either, or, ) `There is either a tiger in this room, or a lady biconditional (equivalence) (if and only if, ) `There is a lady in this room if and only if there is a tiger in the other room
8 Discrete Mathematics Propositional Logic 2-8 Truth Tables Truth table is a way to specify the exact dependence of the truth value of a compound statement through the values of primitive statements involved truth values of primitive statements (propositional variables) p q Φ truth value of compound statements (formulas)
9 Discrete Mathematics Propositional Logic 2-9 Truth Tables of Connectives (Negation and Conjunction) Negation p p F (0) T (1) T (1) F (0) unary connective `Today is Friday `Today is not Friday Conjunction p q p q binary connective `Today is Friday `It is raining `Today is Friday and it is raining
10 Discrete Mathematics Propositional Logic 2-10 Truth Tables of Connectives (Disjunction) Disjunction inclusive `or p q p q `Students who have taken calculus can take this course `Students who have taken computing can take this course `Students who have taken calculus or computing can take this course Be careful with `or constructions in natural languages! `You do your homework, or you ll fail the exam. `Today is Friday or Saturday
11 Discrete Mathematics Propositional Logic 2-11 Truth Tables of Connectives (Exclusive or) Exclusive `or One of the statements is true but not both p q p q `You can follow the rules or be disqualified. `Natalie will arrive today or Natalie will not arrive at all.
12 Discrete Mathematics Propositional Logic 2-12 Truth Tables of Connectives (Implication) Implication p q p q Note that logical (material) implication does not assume any causal connection. `If black is white, then we live in Antarctic. If pigs fly, then Paris is in France.
13 Discrete Mathematics Propositional Logic 2-13 Implication as a Promise Implication can be thought of as a promise, and it is true if the promise is kept `If I am elected, then I will lower taxes - He is not elected and taxes are not lowered promise kept! - He is not elected and taxes are lowered promise kept! - He is elected, but (=and) taxes are not lowered promise broken! - He is elected and taxes are lowered promise kept!
14 Discrete Mathematics Propositional Logic 2-14 Playing with Implication Parts of implication p q hypothesis antecedent premise conclusion consequence `if p, then q `if p, q `p is sufficient for q `q if p `q when p `q unless p `p implies q `p only if q `q whenever p `q follows from p `a sufficient condition for q is p `a necessary condition for p is q
15 Discrete Mathematics Propositional Logic 2-15 Playing with Implication (cntd) Converse, contrapositive, and inverse p q Converse `The home team wins whenever it is raining (`If it is raining then the home team wins ) q p `If the home team wins, then it is raining Contrapositive q p `If the home team does not win, then it is not raining Inverse p q `If it is not raining, then the home team does not win
16 Discrete Mathematics Propositional Logic 2-16 Truth Tables of Connectives (Biconditional) Biconditional or Equivalence One of the statements is true if and only if the other is true p q p q `You can take the flight if and only if you buy a ticket.
17 Discrete Mathematics Propositional Logic 2-17 Example `You can access the Internet from campus if you are a computer science major or if you are not a freshman. p - `you can access the Internet from campus q - `you are a computer science major r - `you are a freshman
18 Discrete Mathematics Propositional Logic 2-18 Homework Exercises from the Book: No. 1,3, 4, 8a, 8c (page 54)
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c Oksana Shatalov, Fall 2017 1 1.1 Language and Logic Mathematical Statements DEFINITION 1. A proposition is any declarative sentence (i.e. it has both a subject and a verb) that is either true or false,