WEEK 1

M: Propositional connectives -- completion. Converse and contraposition. Propositional expressions. Truth tables.
      1.7. Definition. Converse
      1.8. Definition. Contraposition.
      1.9. Informal definition/formal definition. Propositional expression.
      PROBLEMS FOR DISCUSSIONS:  Book, p.7 Exercises 1, 2ac, 3 and p.15 Exercises 15, 18c, 19, 21
W:  Equivalence of propositional expressions. Tautologies, contradictions. Examples. Constant symbols and variables.
       Open sentences. Existential and universal quantifiers.  
      1.10. Definition. Equivalence of two propositional expressions, X <=> Y
      1.11. Definition. Tautology, contradiction.
      1.12. Informal definition. Open sentence.
      1.13. Definition. Existential quantifier, universal quantifier.
      PROBLEMS FOR DISCUSSIONS:  Book, p.16 Exercise 22, 23,31.
               Express using quantifiers: 1. Domain is the set of all real numbers.
                                                                (a) There is a positive real number
                                                                (b) There are two distinct numbers
                                                                (c) There are at least two numbers between 1 and 2
                                                                (d) There is no largest number 
                                                                (e) The equation x²-5x+1  has at most two solutions.
                                                                (f) The equation x²=1 has exactly  two solutions. 
                                                             2. Domain is the set of all positive integers 
                                                                  (a) There is  a smallest number 
                                                                  (b) There is no number between 2 and 3 
                                                                  (c) Number x is a product of two numbers each of them is disctinct from 1 
                                                                  (d) The sum of x,y is a square of some number
                                                             3. The domain comprise all men from the barber's village.
                                                                  (a) The barber shaves precisely all those men in his village who
                                                                        do not shave themselves.
             In each case in 1. and 2. determine whether the assertion is true or false.
             Regarding 3.: Who shaves the barber?
F: Building negations. DeMorgan Laws. Resemblances Existential quantifier -- v and Universal quantifier -- &.
     Basic notions from number theory.
     1.14. Proposition. (a) ¬¬P  <=>  P
                                   (b) ¬ (P & Q)  <=>  ¬P v ¬Q  (De Morgan)
                                   (c) ¬ (P v Q)  <=>  ¬P & ¬Q   (De Morgan)
                                   (d) ¬ (P --> Q) <=>  P & ¬Q
     1.15. Proposition. (a) ¬ ( Ex)P(x) <=> (Ax)¬P(x)
                                   (b) ¬ (Ax)P(x) <=> (Ex)¬P(x)
     1.16. Definition. Divisibility, divisor.
     1.17. Definition. Common divisor. 
     1.18. Definition. Even and odd integer.
     1.19. Definition. Prime.
     1.20. Definition. Relative primes.
     PROBLEMS FOR DISCUSSIONS: Book, p.16 Exercise 25, 26; p.17 Exercise 35, 36, 37, 38, 39;
                                                                            p.23 Exercise 42, 44, 45; p.24 Exercise 48, 50, 52 b,c,e,f