Lets get started with... Logic! Spring 2003 CMSC 203 - Discrete Structures 1 Logic Crucial for mathematical reasoning Used for designing electronic circuitry Logic is a system based on propositions. A proposition is a statement that is either true or false (not both). We say that the truth value of a proposition is either true (T) or false (F).

Corresponds to 1 and 0 in digital circuits Spring 2003 CMSC 203 - Discrete Structures 2 The Statement/Proposition Game Elephants are bigger than mice. Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? true Spring 2003

CMSC 203 - Discrete Structures 3 The Statement/Proposition Game 520 < 111 Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? fals e Spring 2003

CMSC 203 - Discrete Structures 4 The Statement/Proposition Game y > 5 Is this a statement? yes Is this a proposition? no Its truth value depends on the value of y, but this value is not specified. We call this type of statement a propositional function or open sentence . Spring 2003 CMSC 203 - Discrete Structures 5

The Statement/Proposition Game Today is January 1 and 99 < 5. Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? fals e Spring 2003 CMSC 203 - Discrete Structures 6 The Statement/Proposition

Game Please do not fall asleep. Is this a statement? no Its a request. Is this a proposition? no Only statements can be propositions. Spring 2003 CMSC 203 - Discrete Structures 7 The Statement/Proposition Game If elephants were red, they could hide in cherry trees. Is this a statement? yes

Is this a proposition? yes What is the truth value of the proposition? probably false Spring 2003 CMSC 203 - Discrete Structures 8 The Statement/Proposition Game x < y if and only if y > x. Is this a statement? yes Is this a proposition?

yes because its truth value does not depend on specific values of x and y. What is the truth value true ofSpringthe 2003 proposition? CMSC 203 - Discrete Structures 9 Combining Propositions As we have seen in the previous examples, one or more propositions can be combined to form a single compound proposition. We formalize this by denoting propositions with letters such as p, q, r, s, and introducing several logical operators. Spring 2003 CMSC 203 - Discrete Structures

10 Logical Operators (Connectives) We will examine the following logical operators: Negation Conjunction Disjunction Exclusive or Implication Biconditional (NOT) (AND) (OR) (XOR) (if then) (if and only if) Truth tables can be used to show how these operators can combine propositions to compound propositions.

Spring 2003 CMSC 203 - Discrete Structures 11 Negation (NOT) Unary Operator, Symbol: Spring 2003 P P true (T) false (F) false (F) true (T) CMSC 203 - Discrete Structures

12 Conjunction (AND) Binary Operator, Symbol: Spring 2003 P Q P Q T T T T F F

F T F F F F CMSC 203 - Discrete Structures 13 Disjunction (OR) Binary Operator, Symbol: Spring 2003 P Q

P Q T T T T F T F T T F F F

CMSC 203 - Discrete Structures 14 Exclusive Or (XOR) Binary Operator, Symbol: Spring 2003 P Q PQ T T F T F

T F T T F F F CMSC 203 - Discrete Structures 15 Implication (if - then) Binary Operator, Symbol: Spring 2003 P

Q P Q T T T T F F F T T F

F T CMSC 203 - Discrete Structures 16 Biconditional (if and only if) Binary Operator, Symbol: Spring 2003 P Q P Q T T T

T F F F T F F F T CMSC 203 - Discrete Structures 17 Statements and Operators Statements and operators can be combined in

any way to form new statements. P Q P Q (P)(Q) T T F F F T F

F T T F T T F T F F T T

T Spring 2003 CMSC 203 - Discrete Structures 18 Statements and Operations Statements and operators can be combined in any way to form new statements. P Q T T T F

F T F F T T F T F T T F F

F T T Spring 2003 PQ (PQ) (P)(Q) CMSC 203 - Discrete Structures 19 Equivalent Statements P Q (PQ) (P)(Q) (PQ)(P)(Q) T

T F F T T F T T T F T T

T T F F T T T The statements (PQ) and (P) (Q) are logically equivalent, since (PQ) (P) (Q) is always true. Spring 2003 CMSC 203 - Discrete Structures 20 Tautologies and Contradictions A tautology is a statement that is always true.

Examples: R(R) (PQ)(P)(Q) If ST is a tautology, we write ST. If ST is a tautology, we write ST. Spring 2003 CMSC 203 - Discrete Structures 21 Tautologies and Contradictions A contradiction is a statement that is always false. Examples: R(R) ((PQ)(P)(Q)) The negation of any tautology is a contradiction, and the negation of any contradiction is a tautology. Spring 2003 CMSC 203 - Discrete Structures

22 Exercises We already know the following tautology: (PQ) (P)(Q) Nice home exercise: Show that (PQ) (P)(Q). These two tautologies are known as De Morgans laws. Table 5 in Section 1.2 shows many useful laws. Exercises 1 and 7 in Section 1.2 may help you get used to propositions and operators. Spring 2003 CMSC 203 - Discrete Structures 23 Propositional Functions Propositional function (open sentence): statement involving one or more variables, e.g.: x-3 > 5. Let us call this propositional function P(x), where P is the predicate and x is the variable.

What is the truth value of P(2) ? false What is the truth value of P(8) ? false What is the truth value of P(9) ? true Spring 2003 CMSC 203 - Discrete Structures 24 Propositional Functions Let us consider the propositional function Q(x, y, z) defined as: x + y = z. Here, Q is the predicate and x, y, and z are the variables. What is the truth value of Q(2, 3, 5) ? What is the truth value of Q(0, 1, 2) ? What is the truth value of Q(9, -9, 0) ? Spring 2003 CMSC 203 - Discrete Structures

true false true 25 Universal Quantification Let P(x) be a propositional function. Universally quantified sentence: For all x in the universe of discourse P(x) is true. Using the universal quantifier : x P(x) for all x P(x) or for every x P(x) (Note: x P(x) is either true or false, so it is a proposition, not a propositional function.) Spring 2003 CMSC 203 - Discrete Structures 26 Universal Quantification Example: S(x): x is a UMBC student. G(x): x is a genius. What does x (S(x) G(x)) mean ?

If x is a UMBC student, then x is a genius. or All UMBC students are geniuses. Spring 2003 CMSC 203 - Discrete Structures 27 Existential Quantification Existentially quantified sentence: There exists an x in the universe of discourse for which P(x) is true. Using the existential quantifier : x P(x) There is an x such that P(x). There is at least one x such that P(x). (Note: x P(x) is either true or false, so it is a proposition, but no propositional function.) Spring 2003 CMSC 203 - Discrete Structures 28

Existential Quantification Example: P(x): x is a UMBC professor. G(x): x is a genius. What does x (P(x) G(x)) mean ? There is an x such that x is a UMBC professor and x is a genius. or At least one UMBC professor is a genius. Spring 2003 CMSC 203 - Discrete Structures 29 Quantification Another example: Let the universe of discourse be the real numbers. What does xy (x + y = 320) mean ? For every x there exists a y so that x + y = 320. Is it true? yes

Is it true for the natural numbers? no Spring 2003 CMSC 203 - Discrete Structures 30 Disproof by Counterexample A counterexample to x P(x) is an object c so that P(c) is false. Statements such as x (P(x) Q(x)) can be disproved by simply providing a counterexample. Statement: All birds can fly. Disproved by counterexample: Penguin. Spring 2003 CMSC 203 - Discrete Structures

31 Negation (x P(x)) is logically equivalent to x (P(x)). (x P(x)) is logically equivalent to x (P(x)). See Table 3 in Section 1.3. I recommend exercises 5 and 9 in Section 1.3. Spring 2003 CMSC 203 - Discrete Structures 32 Logical Equivalences Identity Laws: p T p and p F p. Domination Laws: p T T and p F F. Idempotent Laws: p p p and p p p. Double Negation Law: ( p) p.

Commutative Laws: (p q) (q p) and (p q) (q p). Associative Laws: (p q) r p (q r) and (p q) r p (q r). Spring 2003 CMSC 203 - Discrete Structures 33 Logical Equivalences Distributive Laws: p (q r) (p q) (p r) and p (q r) (p q) (p r). DeMorgans Laws: (p q) ( p q) and (p q) ( p q). Absorption Laws: p (p q) p and p (p q) p. Negation Laws: p p T and p p F. Spring 2003 CMSC 203 - Discrete Structures

34 Examples Find the truth table of [p (q r)]. (An important Theorem) Show that: p q p q. Show the Corollary: (p q) p q. Using the tables on page 24, verify the Absorption Laws: p (p q) p, and p (p q) p. Spring 2003 CMSC 203 - Discrete Structures 35 Examples Negate: (a) For each integer, n, if 4 divides n, then 2 divides n. (b) xR, yR [x y] [y (x + 1)]. Spring 2003

CMSC 203 - Discrete Structures 36