Spring 2003CMSC Discrete Structures1 Let’s get started with... Logic !

Spring 2003CMSC Discrete Structures2 Logic Crucial for mathematical reasoningCrucial for mathematical reasoning Used for designing electronic circuitryUsed for designing electronic circuitry Logic is a system based on propositions.Logic is a system based on propositions. A proposition is a statement that is either true or false (not both).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).We say that the truth value of a proposition is either true (T) or false (F). Corresponds to 1 and 0 in digital circuitsCorresponds to 1 and 0 in digital circuits

Spring 2003CMSC Discrete Structures3 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 2003CMSC Discrete Structures4 The Statement/Proposition Game “520 < 111” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? false

Spring 2003CMSC Discrete Structures5 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 2003CMSC Discrete Structures6 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? false

Spring 2003CMSC Discrete Structures7 The Statement/Proposition Game “Please do not fall asleep.” Is this a statement? no Is this a proposition? no Only statements can be propositions. It’s a request.

Spring 2003CMSC Discrete Structures8 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 2003CMSC Discrete Structures9 The Statement/Proposition Game “x x.” Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? true … because its truth value does not depend on specific values of x and y.

Spring 2003CMSC Discrete Structures10 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 2003CMSC Discrete Structures11 Logical Operators (Connectives) We will examine the following logical operators: Negation (NOT) Negation (NOT) Conjunction (AND) Conjunction (AND) Disjunction (OR) Disjunction (OR) Exclusive or (XOR) Exclusive or (XOR) Implication (if – then) Implication (if – then) Biconditional (if and only if) Biconditional (if and only if) Truth tables can be used to show how these operators can combine propositions to compound propositions.

Spring 2003CMSC Discrete Structures12 Negation (NOT) Unary Operator, Symbol:  P PPPP true (T) false (F) true (T)

Spring 2003CMSC Discrete Structures13 Conjunction (AND) Binary Operator, Symbol:  PQ PQPQPQPQ TTT TFF FTF FFF

Spring 2003CMSC Discrete Structures14 Disjunction (OR) Binary Operator, Symbol:  PQ PQPQPQPQ TTT TFT FTT FFF

Spring 2003CMSC Discrete Structures15 Exclusive Or (XOR) Binary Operator, Symbol:  PQ PQPQPQPQ TTF TFT FTT FFF

Spring 2003CMSC Discrete Structures16 Implication (if - then) Binary Operator, Symbol:  PQ PQPQPQPQ TTT TFF FTT FFT

Spring 2003CMSC Discrete Structures17 Biconditional (if and only if) Binary Operator, Symbol:  PQ PQPQPQPQ TTT TFF FTF FFT

Spring 2003CMSC Discrete Structures18 Statements and Operators Statements and operators can be combined in any way to form new statements. PQ PPPP QQQQ (  P)  (  Q) TTFFF TFFTT FTTFT FFTTT

Spring 2003CMSC Discrete Structures19 Statements and Operations Statements and operators can be combined in any way to form new statements. PQ PQPQPQPQ  (P  Q) (  P)  (  Q) TTTFF TFFTT FTFTT FFFTT

Spring 2003CMSC Discrete Structures20 Equivalent Statements PQ  (P  Q) (  P)  (  Q)  (P  Q)  (  P)  (  Q) TTFFT TFTTT FTTTT FFTTT The statements  (P  Q) and (  P)  (  Q) are logically equivalent, since  (P  Q)  (  P)  (  Q) is always true.

Spring 2003CMSC Discrete Structures21 Tautologies and Contradictions A tautology is a statement that is always true. Examples: R  (  R)R  (  R)  (P  Q)  (  P)  (  Q)  (P  Q)  (  P)  (  Q) If S  T is a tautology, we write S  T. If S  T is a tautology, we write S  T.

Spring 2003CMSC Discrete Structures22 Tautologies and Contradictions A contradiction is a statement that is always false.Examples: R  (  R)R  (  R)  (  (P  Q)  (  P)  (  Q))  (  (P  Q)  (  P)  (  Q)) The negation of any tautology is a contra- diction, and the negation of any contradiction is a tautology.

Spring 2003CMSC Discrete Structures23 Exercises We already know the following tautology:  (P  Q)  (  P)  (  Q) Nice home exercise: Show that  (P  Q)  (  P)  (  Q). These two tautologies are known as De Morgan’s 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 2003CMSC Discrete Structures24 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) ? What is the truth value of P(9) ? false true

Spring 2003CMSC Discrete Structures25 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) ? true What is the truth value of Q(0, 1, 2) ? What is the truth value of Q(9, -9, 0) ? false true

Spring 2003CMSC Discrete Structures26 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 2003CMSC Discrete Structures27 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 2003CMSC Discrete Structures28 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).” “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 2003CMSC Discrete Structures29 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 2003CMSC Discrete Structures30 Quantification Another example: Let the universe of discourse be the real numbers. What does  x  y (x + y = 320) mean ? “For every x there exists a y so that x + y = 320.” Is it true? Is it true for the natural numbers? yes no

Spring 2003CMSC Discrete Structures31 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 2003CMSC Discrete Structures32 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 2003CMSC Discrete Structures33 Logical Equivalences Identity Laws: p  T  p and p  F  p.Identity Laws: p  T  p and p  F  p. Domination Laws: p  T  T and p  F  F.Domination Laws: p  T  T and p  F  F. Idempotent Laws: p  p  p and p  p  p.Idempotent Laws: p  p  p and p  p  p. Double Negation Law:  (  p)  p.Double Negation Law:  (  p)  p. Commutative Laws:Commutative Laws: (p  q)  (q  p) and (p  q)  (q  p). Associative Laws: (p  q)  r  p  (q  r)Associative Laws: (p  q)  r  p  (q  r) and (p  q)  r  p  (q  r).

Spring 2003CMSC Discrete Structures34 Logical Equivalences Distributive Laws:Distributive Laws: p  (q  r)  (p  q)  (p  r) and p  (q  r)  (p  q)  (p  r). DeMorgan’s Laws:DeMorgan’s Laws:  (p  q)  (  p   q) and  (p  q)  (  p   q). Absorption Laws:Absorption Laws: p  (p  q)  p and p  (p  q)  p. Negation Laws: p   p  T and p   p  F.Negation Laws: p   p  T and p   p  F.

Spring 2003CMSC Discrete Structures35 Examples Find the truth table of  [p  (q  r)].Find the truth table of  [p  (q  r)]. (An important Theorem) Show that:(An important Theorem) Show that: p  q   p  q. Show the Corollary:  (p  q)  p   q.Show the Corollary:  (p  q)  p   q. Using the tables on page 24, verify the Absorption Laws:Using the tables on page 24, verify the Absorption Laws: p  (p  q)  p, and p  (p  q)  p.

Spring 2003CMSC Discrete Structures36 Examples Negate:Negate: (a) For each integer, n, if 4 divides n, then 2 divides n. (b)  x  R,  y  R  [x  y]  [y  (x + 1)].