1. Introduction to Logic & Set Theory Martin Russell EE1J2 Discrete Mathematics

2. EE1J2 - Slide 2 Outline of lecture 1 Introduction to course Textbooks What is formal logic? Propositional logic Propositions Propositional connectives Formalisation of arguments

3. EE1J2 - Slide 3 EE1J2 Propositional Logic Basic ideas: propositions, connectives Formalising statements in ‘natural language’ Formal proofs Set theory Basic ideas: definitions of sets Relations, functions and equivalence relations Cardinality, finite, countable and uncountable sets Predicate Logic Logic programming

4. EE1J2 - Slide 4 Tutorial sheets One tutorial sheet handed out each week To be handed in approx 1 week later Questions and model solutions will appear on my web pages

5. EE1J2 - Slide 5 Web Pages http://www.eee.bham.ac.uk/russellm/ee1j2.html All materials used in course will appear on course web pages Powerpoint slides Homework sheets Homework solutions Please don’t print the notes – I’ll hand out copies each week

6. EE1J2 - Slide 6 Class test (week 6) Questions in class test will be selected from tutorial sheets So, if you do the tutorial sheets you should do well in the class test!

7. EE1J2 - Slide 7 Recommended Books John Truss, “Discrete Mathematics for Computer Scientists”, Addison-Wesley, Second Edition, 1999

8. EE1J2 - Slide 8 Recommended Books (cont.) Nimal Nissanke, “Introductory Logic and Sets for Computer Scientists”, Addison-Wesley, 1999

9. EE1J2 - Slide 9 Recommended Books New this year: James A Anderson, “Discrete Mathematics with Combinatorics (2 nd edition)”, Prentice- Hall, 2004

10. EE1J2 - Slide 10 Recommended books (cont.) Rod Haggarty “Discrete mathematics for computing” Addison Wesley A Chetwynd and P Diggle, “Discrete Mathematics” Butterworth- Heinemann

11. EE1J2 - Slide 11 Other books I have also made use of two other books: Geoffrey Finch, “How to study linguistics”, Macmillan, 1998 J N Crossley and others, “What is mathematical logic?”, Oxford University Press, 1972

12. EE1J2 - Slide 12 Introduction to logic What is logic? Why is it useful? Types of logic Propositional logic Predicate logic

13. EE1J2 - Slide 13 What is logic? “Logic is the beginning of wisdom, not the end”

14. EE1J2 - Slide 14 What is logic? Logic n.1. The branch of philosophy concerned with analysing the patterns of reasoning by which a conclusion is drawn from a set of premises, without reference to meaning or context (Collins English Dictionary)

15. EE1J2 - Slide 15 Why study logic? Logic is concerned with two key skills, which any computer engineer or scientist should have: Abstraction Formalisation

16. EE1J2 - Slide 16 Why is logic important? Logic is a formalisation of reasoning. Logic is a formal language for deducing knowledge from a small number of explicitly stated premises (or hypotheses, axioms, facts) Logic provides a formal framework for representing knowledge Logic differentiates between the structure and content of an argument

17. EE1J2 - Slide 17 Logic as formal language In this course, logic will be presented as a formal language Within that formal language: Knowledge can be stated concisely and precisely The process of reasoning from that knowledge can be made rigorous

18. EE1J2 - Slide 18 What is an argument? An argument is just a sequence of statements. Some of these statements, the premises, are assumed to be true and serve as a basis for accepting another statement of the argument, called the conclusion

19. EE1J2 - Slide 19 Deduction and inference If the conclusion is justified, based solely on the premises, the process of reasoning is called deduction If the validity of the conclusion is based on generalisation from the premises, based on strong but inconclusive evidence, the process is called inference (sometimes called induction) This course is concerned only with deduction

20. EE1J2 - Slide 20 Two examples Deductive argument: “Alexandria is a port or a holiday resort. Alexandria is not a port. Therefore, Alexandria is a holiday resort” Inductive argument “Most students who did not do the tutorial questions will fail the exam. John did not do the tutorial questions. Therefore John will fail the exam”

21. EE1J2 - Slide 21 Some different types of logic Historically, a number of types of logic have been proposed. In this course we will study Propositional logic (Boole, 1815-1864) Predicate logic (Frege 1848-1925)

22. EE1J2 - Slide 22 Propositional logic Simple types of statements, called propositions, are treated as atomic building blocks for more complex statements Alexandria is a port or a holiday resort. Alexandria is not a port. Therefore, Alexandria is a holiday resort

23. EE1J2 - Slide 23 Propositional logic Basic propositions in the argument are p – Alexandria is a port q – Alexandria is a holiday resort. In abstract form, the argument becomes p or q Not q Therefore q

24. EE1J2 - Slide 24 Predicate logic Extension of propositional logic A ‘predicate’ is just a property Predicates define relationships between any number of entities using qualifiers:  “for all”, “for every”  “there exists”

25. EE1J2 - Slide 25 Example Let P(x) be the property ‘if x is a triangle then the sum of its internal angles is 180 o ” In predicate logic:  x P(x) “For every x such that x is a triangle, the sum of the internal angles of x is 180 o”

26. EE1J2 - Slide 26 Another example Let P(x) be the property ‘x is an integer and x 2 = 4’ Then  x P(x) “There exists x such that x is an integer and x 2 = 4”

27. EE1J2 - Slide 27 Newton’s second law of motion  x: Object  stationary(x)  in-uniform-motion (x)   f : Force  x is-acted-upon-by f In English: “for every x of a certain type referred to as an Object, x is stationary, x is in uniform motion, or there is an f of type Force such that x is acted upon by f” “for every x”“of type called object”“or” “there exists an f”

28. EE1J2 - Slide 28  and  Remember:  x ‘for every x’, or ‘for All x’  x ‘there is an x’ or ‘there Exists an x’ Tip: Think of  as an upside down ‘A’ (‘for All’) Think of  as a backwards ‘E’ (‘there Exists’)

29. EE1J2 - Slide 29 Propositions A (atomic, elementary) proposition is the underlying meaning of a simple declarative sentence, which is either true or false The truth or falsehood of a proposition is indicated by assigning it one of the truth values T (for true) or F (for false)

30. EE1J2 - Slide 30 Example propositions Mammals are warm blooded The sun orbits the earth The evergreen forests of Canada consist of spruce, pine and fir trees John is taller than Joan Joan is shorter than John John is not shorter than Joan

31. EE1J2 - Slide 31 Sentences which are not propositions Over millions of years they build up on top of one another to form a reef Can the arctic hare change the colour of its coat to match its surroundings? Put down that book!

32. EE1J2 - Slide 32 Which are propositions? Can pigs fly? Pigs can fly Sparrows can fly Joe runs faster than Patrick Patrick runs slower than Joe Pay your bills on time The circumference of a circle is equal to four times its diameter

33. EE1J2 - Slide 33 Propositional connectives These are the words that we use to join atomic propositions together to form compound propositions. E.G: In 1938 Hitler seized Austria, (and) in 1939 he seized former Czechoslovakia and in 1941 he attacked the former USSR while still having a non-aggression pact with it

34. EE1J2 - Slide 34 Propositional connectives Propositional logic has four connectives NameRead asSymbol negation‘not’  conjunction‘and’  disjunction‘or’  implication‘if…then…’ 

35. EE1J2 - Slide 35 ConnectiveInterpretation  negation  p is true if and only if p is false  A conjunction p  q is true if and only if both p and q are true  A disjunction p  q is true if and only if p is true or q is true.  An implication p  q is false if and only if p is true and q is false Interpretation of connectives

36. EE1J2 - Slide 36 Some more terminology… Expressions either side of a conjunction are called conjuncts (p  q) Expressions either side of a disjunction are called disjuncts (p  q) In the implication p  q, p is called the antecedent and q is the consequence

37. EE1J2 - Slide 37 Precedence of connectives In complex propositions, brackets may be used to remove ambiguity. (p  q)  r versus p  (q  r) By convention, the order of precedence Brackets, Negation, Conjunction, Disjunction, Implication

38. EE1J2 - Slide 38 Formalisation Statement In 1938 Hitler seized Austria, (and) in 1939 he seized former Czechoslovakia and in 1941 he attacked the former USSR while still having a non-aggression pact with it

39. EE1J2 - Slide 39 Formalisation (continued) Atomic propositions: p – In 1938 Hitler seized Austria q – In 1939 Hitler seized former Czechoslovakia r – In 1941 Hitler attacked the former USSR s – In 1941 Hitler had a non-aggression pact with the former USSR Formalisation in Propositional Logic: p  q  r  s

40. EE1J2 - Slide 40 Although both Stanley and Gordon are not young, Stanley has a better chance of winning the next bowling tournament, despite Gordon’s considerable experience Formalisation

41. EE1J2 - Slide 41 Formalisation (continued) Atomic propositions: p – Stanley is young q – Gordon is young r – Stanley has a better chance of winning the next bowling tournament s – Gordon has considerable experience in bowling Formalisation in Propositional Logic: (  p)  (  q)  r  s

42. EE1J2 - Slide 42 Negation and atomic propositions Note that for first atomic proposition I chose: Stanley is young and not Stanley is not young

43. EE1J2 - Slide 43 Summary of Lecture 1 Introduction to course Textbooks What is formal logic? Propositional logic Propositions Propositional connectives Formalisation of arguments