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Let’s get started with with discrete math.pptx
- 1. Let’s get startedwith... Logic! Fall 2002 CMSC 203 - Discrete Structures 1
- 2. LOGIC Let’s get startedwith... LOGIC Fall 2002 CMSC 203 - Discrete Structures 2
- 3. Logic • Crucial formathematical 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 Fall 2002 CMSC 203 - Discrete Structures 3
- 4. The Statement/Proposition Game “Elephantsare bigger than mice.” Fall 2002 CMSC 203 - Discrete Structures 4 Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? true
- 5. The Statement/Proposition Game “520< 111” Fall 2002 CMSC 203 - Discrete Structures 5 Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? false
- 6. The Statement/Proposition Game “y> 5” Fall 2002 CMSC 203 - Discrete Structures 6 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.
- 7. The Statement/Proposition Game “Todayis January 1 and 99 < 5.” Fall 2002 CMSC 203 - Discrete Structures 7 Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? false
- 8. The Statement/Proposition Game “Pleasedo not fall asleep.” Fall 2002 CMSC 203 - Discrete Structures 8 Is this a statement? no Is this a proposition? no Only statements can be propositions. It’s a request.
- 9. The Statement/Proposition Game “Ifelephants were red, they could hide in cherry trees.” Fall 2002 CMSC 203 - Discrete Structures 9 Is this a statement? yes Is this a proposition? yes What is the truth value of the proposition? probably false
- 10. The Statement/Proposition Game “x< y if and only if y > x.” Fall 2002 CMSC 203 - Discrete Structures 10 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.
- 11. Combining Propositions As wehave 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. Fall 2002 CMSC 203 - Discrete Structures 11
- 12. Logical Operators (Connectives) Wewill examine the following logical operators: • Negation (NOT) • Conjunction (AND) • Disjunction (OR) • Exclusive or (XOR) • Implication (if – then) • Biconditional (if and only if) Truth tables can be used to show how these operators can combine propositions to compound propositions. Fall 2002 CMSC 203 - Discrete Structures 12
- 13. Negation (NOT) Unary Operator,Symbol: Fall 2002 CMSC 203 - Discrete Structures 13 P P true (T) false (F) false (F) true (T)
- 14. Conjunction (AND) Binary Operator,Symbol: Fall 2002 CMSC 203 - Discrete Structures 14 P Q PQ T T T T F F F T F F F F
- 15. Disjunction (OR) Binary Operator,Symbol: Fall 2002 CMSC 203 - Discrete Structures 15 P Q PQ T T T T F T F T T F F F
- 16. Exclusive Or (XOR) BinaryOperator, Symbol: Fall 2002 CMSC 203 - Discrete Structures 16 P Q PQ T T F T F T F T T F F F
- 17. Implication (if -then) Binary Operator, Symbol: Fall 2002 CMSC 203 - Discrete Structures 17 P Q PQ T T T T F F F T T F F T
- 18. Biconditional (if andonly if) Binary Operator, Symbol: Fall 2002 CMSC 203 - Discrete Structures 18 P Q PQ T T T T F F F T F F F T
- 19. Statements and Operators Statements and operators can be combined in any way to form new statements. Fall 2002 CMSC 203 - Discrete Structures 19 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
- 20. Statements and Operations Statements and operators can be combined in any way to form new statements. Fall 2002 CMSC 203 - Discrete Structures 20 P Q PQ (PQ) (P)(Q) T T T F F T F F T T F T F T T F F F T T
- 21. Equivalent Statements Thestatements (PQ) and (P) (Q) are logically equivalent, since (PQ) (P) (Q) is always true. Fall 2002 CMSC 203 - Discrete Structures 21 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
- 22. Tautologies and Contradictions Atautology 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. Fall 2002 CMSC 203 - Discrete Structures 22
- 23. Tautologies and Contradictions Acontradiction is a statement that is always false. Examples: • R(R) • ((PQ)(P)(Q)) The negation of any tautology is a contra- diction, and the negation of any contradiction is a tautology. Fall 2002 CMSC 203 - Discrete Structures 23
- 24. Exercises We already knowthe following tautology: (PQ) (P)(Q) Nice home exercise: Show that (PQ) (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. Fall 2002 CMSC 203 - Discrete Structures 24
- 25. Let’s Talk AboutLogic • 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 Fall 2002 CMSC 203 - Discrete Structures 25
- 26. Logical Operators (Connectives) •Negation (NOT) • Conjunction (AND) • Disjunction (OR) • Exclusive or (XOR) • Implication (if – then) • Biconditional (if and only if) Truth tables can be used to show how these operators can combine propositions to compound propositions. Fall 2002 CMSC 203 - Discrete Structures 26
- 27. Tautologies and Contradictions Atautology 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. Fall 2002 CMSC 203 - Discrete Structures 27
- 28. Tautologies and Contradictions Acontradiction 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. Fall 2002 CMSC 203 - Discrete Structures 28
- 29. 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. Fall 2002 CMSC 203 - Discrete Structures 29 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
- 30. Propositional Functions Let usconsider 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. Fall 2002 CMSC 203 - Discrete Structures 30 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
- 31. 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.) Fall 2002 CMSC 203 - Discrete Structures 31
- 32. Universal Quantification Example: S(x): xis 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.” Fall 2002 CMSC 203 - Discrete Structures 32
- 33. Existential Quantification Existentially quantifiedsentence: 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.) Fall 2002 CMSC 203 - Discrete Structures 33
- 34. Example: P(x): x isa 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.” Fall 2002 CMSC 203 - Discrete Structures 34 Existential Quantification
- 35. Quantification Another example: Let theuniverse 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.” Fall 2002 CMSC 203 - Discrete Structures 35 Is it true? Is it true for the natural numbers? yes no
- 36. Disproof by Counterexample Acounterexample 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. Fall 2002 CMSC 203 - Discrete Structures 36 Statement: “All birds can fly.” Disproved by counterexample: Penguin.
- 37. Negation (x P(x)) islogically equivalent to x (P(x)). (x P(x)) is logically equivalent to x (P(x)). Fall 2002 CMSC 203 - Discrete Structures 37
- 38. … and nowfor something completely different… Set Theory Fall 2002 CMSC 203 - Discrete Structures 38 Actually, you will see that logic and set theory are very closely related.
- 39. Set Theory • Set:Collection of objects (“elements”) • aA “a is an element of A” “a is a member of A” • aA “a is not an element of A” • A = {a1, a2, …, an} “A contains…” • Order of elements is meaningless • It does not matter how often the same element is listed. Fall 2002 CMSC 203 - Discrete Structures 39
- 40. Set Equality Sets Aand B are equal if and only if they contain exactly the same elements. Examples: Fall 2002 CMSC 203 - Discrete Structures 40 • A = {9, 2, 7, -3}, B = {7, 9, -3, 2} : A = B • A = {dog, cat, horse}, B = {cat, horse, squirrel, dog} : A B • A = {dog, cat, horse}, B = {cat, horse, dog, dog} : A = B
- 41. Examples for Sets “Standard” Sets: • Natural numbers N = {0, 1, 2, 3, …} • Integers Z = {…, -2, -1, 0, 1, 2, …} • Positive Integers Z+ = {1, 2, 3, 4, …} • Real Numbers R = {47.3, -12, , …} • Rational Numbers Q = {1.5, 2.6, -3.8, 15, …} (correct definition will follow) Fall 2002 CMSC 203 - Discrete Structures 41
- 42. Examples for Sets •A = “empty set/null set” • A = {z} Note: zA, but z {z} • A = {{b, c}, {c, x, d}} • A = {{x, y}} Note: {x, y} A, but {x, y} {{x, y}} • A = {x | P(x)} “set of all x such that P(x)” • A = {x | xN x > 7} = {8, 9, 10, …} “set builder notation” Fall 2002 CMSC 203 - Discrete Structures 42
- 43. Examples for Sets Weare now able to define the set of rational numbers Q: Q = {a/b | aZ bZ+ } or Q = {a/b | aZ bZ b0} And how about the set of real numbers R? R = {r | r is a real number} That is the best we can do. Fall 2002 CMSC 203 - Discrete Structures 43
- 44. Subsets A B“A is a subset of B” A B if and only if every element of A is also an element of B. We can completely formalize this: A B x (xA xB) Examples: Fall 2002 CMSC 203 - Discrete Structures 44 A = {3, 9}, B = {5, 9, 1, 3}, A B ? true A = {3, 3, 3, 9}, B = {5, 9, 1, 3}, A B ? false true A = {1, 2, 3}, B = {2, 3, 4}, A B ?
- 45. Subsets Useful rules: • A= B (A B) (B A) • (A B) (B C) A C (see Venn Diagram) Fall 2002 CMSC 203 - Discrete Structures 45 U A B C
- 46. Useful rules: • A for any set A • A A for any set A Proper subsets: A B “A is a proper subset of B” A B x (xA xB) x (xB xA) or A B x (xA xB) x (xB xA) Fall 2002 CMSC 203 - Discrete Structures 46
- 47. Cardinality of Sets Ifa set S contains n distinct elements, nN, we call S a finite set with cardinality n. Examples: A = {Mercedes, BMW, Porsche}, |A| = 3 Fall 2002 CMSC 203 - Discrete Structures 47 B = {1, {2, 3}, {4, 5}, 6} |B| = 4 C = |C| = 0 D = { xN | x 7000 } |D| = 7001 E = { xN | x 7000 } E is infinite!