Section 1.1: Propositions and Connectives

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Esmond Blair
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1 Section 1.1: Propositions and Connectives Definition : Proposition: A statement that has exactly one truth value, i.e., either true (T) or false (F). Axiom (or Postulate): A statement about the primitive terms that are deliberately chosen as unproved. Theorem: A theorem is statements that are logically deduced from axioms and previously established statements. Examples of propositions: 1. 7 is a prime number. 2. Both 7 and 27 are prime numbers. 3. Either London or Paris is in France. 4. If m is an even number, then m 2 is also an even number. 5. If both m and n are prime numbers, then mn + 1 is a prime number. 6. It is not the case that 2 is a rational number Someone called me yesterday. 8. The ice cream cone sells for 99 cents each in that MacDonald. Examples of statements which are NOT propositions: 1. Do you know UTSA has more than 30,000 students? 2. Yesterday it was so hot. 3. This textbook is very expensive. 4. How do you fix the window? 5. The ice cream in that MacDonald tastes really good. [1/11]- 1.1-FOM

2 Example: Five Postulates (Axioms) about Euclidean geometry Excerpt from: Foundations and Fundamental Concepts of Mathematics, by Edward Eves. [2/11]- 1.1-FOM

3 Example: Axioms about natural numbers Excerpt from: Foundations and Fundamental Concepts of Mathematics, by Edward Eves. [3/11]- 1.1-FOM

4 Examples of theorems: Theorem: 2 is an irrational number. Theorem: The sum of three internal angles of a triangle is 180 degree. Pythagorean Theorem: Let a b c be the lengths of three sides of a triangle. Then the triangle is a right triangle if and only if a 2 + b 2 = c 2. Intermediate Value Theorem: If function f is continuous on [a, b], then for any w in between f(a) and f(b), there exists a number c in between a and b, such that f(c) = w. Fermat s last Theorem: For any integer n 3, equation x n + y n = z n does not have positive integer solutions. (Proved in 1994) Goldbach s conjecture: Every even integer greater than 2 can be expressed as the sum of two primes. (not a theorem, no proof yet.) [4/11]- 1.1-FOM

5 Definition : Let P and Q are given propositions., Conjunction: The conjunction of P and Q, denoted by P Q, is the proposition P and Q. P Q is true only when both P and Q are true, Disjunction: The disjunction of P and Q, denoted by P Q, is the proposition P or Q. P Q is false only when both P Q are false. Negation: The negation of a proposition P, denoted by P, is the proposition not P. P is true when P is false, and false when P is true. Truth Table P Q P Q P Q P T T T T F F T F T T T F F T F F F F F T Truth table for P (Q R): P Q R Q R (Q R) P (Q R) T T T T F F F T T T F F T F T T F F F F T T F F T T F T F F F T F T F F T F F F T T F F F F T F Note: A truth table involving 4 propositions has 2 4 = 16 rows. To build the table systematically, try to vary P (1th column) as T, F, T, F,, then Q (2nd column) as 2 T s followed by 2 F s followed by 2 T s, etc, then to vary R (3rd column) as 4 T s followed by 4 F s, etc.. [5/11]- 1.1-FOM

6 Definition : (Tautology and Contradiction) A tautology is a propositional form (logical formula) that is true for every assignment of truth values to its components. A contradiction is a propositional form (logical formula) that is false for every assignment of truth values to its components. P P is a tautology. P P is a contradiction. (P Q) ( P Q) is a tautology. P Q P Q P Q (P Q) ( P Q) T T T F T F T T F T T F T F T F F F T T [6/11]- 1.1-FOM

7 Definition : (Equivalence) Two propositional forms P and Q are called logically equivalent, if they have the same truth values for evey assignment of its components. Denial of a proposition: if a proposition Q is logically equivalent to the negation of P, then we way Q is a denial of P. P is equiv. to ( P ), and P is a denial of P. P is equiv to ( ( ( P ))), and ( ( P )) is a denial of P. (P Q) is equiv to P Q, and P Q is a denial of P Q. (P Q) is equiv to P Q. and P Q is a denial of P Q. P (P Q) is equiv to P. Show that (P Q) (P R) is equiv to P (P Q). [7/11]- 1.1-FOM

8 Theorem 1 (Basic laws) (a) P equiv to ( P ) double negation law (b) P Q equiv to Q P commutative law (c) P Q equiv to Q P commutative law (d) P (Q R) equiv to (P Q) R associative law (e) P (Q R) equiv to (P Q) R associative law (f) P (Q R) equiv to (P Q) (P R) distributive law (g) P (Q R) equiv to (P Q) (P R) distributive law (h) (P Q) equiv to ( P ) ( Q) DeMorgan s law (i) (P Q) equiv to ( P ) ( Q) DeMorgan s law Remark: (i) Each of P, Q, R in the above laws can be substituted by any propositional forms. (ii) If and are interchanged thorough out of a law, then the equivalence still holds. [8/11]- 1.1-FOM

9 Translate a statement or statements into propositional forms Both 7 and 27 are prime numbers. 2 is an even number and also a prime. It is not the case that π is integer or smaller than 3. Although 11 divides 120, it is neither prime nor a divisor of 100. Both 10 and 20 are integers, but none of them has a factor 3. [9/11]- 1.1-FOM

10 Use logical forms and laws to solve the following puzzle. [10/11]- 1.1-FOM

11 Conventions in applying logical connectives: First, then, and last. P Q R equiv to [ ( P )] [( Q) ( R)] P Q P Q R equiv to ( P ) {[Q ( ( P ))] Q} R [11/11]- 1.1-FOM

2.2: Logical Equivalence: The Laws of Logic

2.2: Logical Equivalence: The Laws of Logic Example (2.7) For primitive statement p and q, construct a truth table for each of the following compound statements. a) p q b) p q Here we see that the corresponding truth tables for two statement p q