This is one of the subtopics of Introduction to set and logic theory in mathematics
- 1. Fall 2002 CMSC 203 - Discrete Structures 1 Let’s get started with... Logic!
- 2. Fall 2002 CMSC 203 - Discrete Structures 2 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
- 3. Fall 2002 CMSC 203 - Discrete Structures 3 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
- 4. Fall 2002 CMSC 203 - Discrete Structures 4 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
- 5. Fall 2002 CMSC 203 - Discrete Structures 5 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.
- 6. Fall 2002 CMSC 203 - Discrete Structures 6 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
- 7. Fall 2002 CMSC 203 - Discrete Structures 7 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.
- 8. Fall 2002 CMSC 203 - Discrete Structures 8 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
- 9. Fall 2002 CMSC 203 - Discrete Structures 9 The Statement/Proposition Game “x < y if and only if y > 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.
- 10. Fall 2002 CMSC 203 - Discrete Structures 10 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.
- 11. Fall 2002 CMSC 203 - Discrete Structures 11 Logical Operators (Connectives) We will 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.
- 12. Fall 2002 CMSC 203 - Discrete Structures 12 Negation (NOT) Unary Operator, Symbol: P P true (T) false (F) false (F) true (T)
- 13. Fall 2002 CMSC 203 - Discrete Structures 13 Conjunction (AND) Binary Operator, Symbol: P Q PQ T T T T F F F T F F F F
- 14. Fall 2002 CMSC 203 - Discrete Structures 14 Disjunction (OR) Binary Operator, Symbol: P Q PQ T T T T F T F T T F F F
- 15. Fall 2002 CMSC 203 - Discrete Structures 15 Exclusive Or (XOR) Binary Operator, Symbol: P Q PQ T T F T F T F T T F F F
- 16. Fall 2002 CMSC 203 - Discrete Structures 16 Implication (if - then) Binary Operator, Symbol: P Q PQ T T T T F F F T T F F T
- 17. Fall 2002 CMSC 203 - Discrete Structures 17 Biconditional (if and only if) Binary Operator, Symbol: P Q PQ T T T T F F F T F F F T
- 18. Fall 2002 CMSC 203 - Discrete Structures 18 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
- 19. Fall 2002 CMSC 203 - Discrete Structures 19 Statements and Operations Statements and operators can be combined in any way to form new statements. 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
- 20. Fall 2002 CMSC 203 - Discrete Structures 20 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.
- 21. Fall 2002 CMSC 203 - Discrete Structures 21 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.
- 22. Fall 2002 CMSC 203 - Discrete Structures 22 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 contra- diction, and the negation of any contradiction is a tautology.
- 23. Fall 2002 CMSC 203 - Discrete Structures 23 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 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.
- 24. Fall 2002 CMSC 203 - Discrete Structures 24 Let’s Talk About Logic • 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
- 25. Fall 2002 CMSC 203 - Discrete Structures 25 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.
- 26. Fall 2002 CMSC 203 - Discrete Structures 26 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.
- 27. Fall 2002 CMSC 203 - Discrete Structures 27 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.