![12
Conjunction, Disjunction
• Conjunction: p q [“and”]
• Disjunction: p q [“or”]
p q p q p q
T T T T
T F F T
F T F T
F F F F](https://image.slidesharecdn.com/1019lec1-221231071722-abddc845/85/1019Lec1-ppt-12-320.jpg)
![15
Conditional
• p q [“if p then q”]
• p: hypothesis, q: conclusion
• E.g.: “If you turn in a homework late, it will not
be graded”; “If you get 100% in this course,
you will get an A+”.
• TRICKY: Is p q TRUE if p is FALSE?
YES!!
• Think of “If you get 100% in this course, you
will get an A+” as a promise – is the promise
violated if someone gets 50% and does not
receive an A+?](https://image.slidesharecdn.com/1019lec1-221231071722-abddc845/85/1019Lec1-ppt-15-320.jpg)
![16
Conditional - 2
• p q [“if p then q”]
• Truth table:
p q p q p q
T T T T
T F F F
F T T T
F F T T
Note the truth table of p q](https://image.slidesharecdn.com/1019lec1-221231071722-abddc845/85/1019Lec1-ppt-16-320.jpg)
1019Lec1.ppt
- 1. 1 Math/CSE 1019C: Discrete Mathematics for Computer Science Fall 2011 Suprakash Datta datta@cse.yorku.ca Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cse.yorku.ca/course/1019
- 2. 2 Administrivia Kenneth H. Rosen. Discrete Mathematics and Its Applications, 7th Edition. McGraw Hill, 2012. Lectures: Tu-Th 10:00-11:30 am (CLH E) Exams: 3 tests (45%), final (40%) Homework (15%): equally divided between several assignments. Slides: should be available after the class Office hours: Wed 3-5 pm or by appointment at CSEB 3043. Textbook:
- 3. 3 Administrivia – contd. • Cheating will not be tolerated. Visit the class webpage for more details on policies. • TA: Tutorials/office hours TBA. • HW submitted late will not be graded.
- 4. 4 Course objectives We will focus on two major goals: • Basic tools and techniques in discrete mathematics – Propositional logic – Set Theory – Simple algorithms – Induction, recursion – Counting techniques (Combinatorics) • Precise and rigorous mathematical reasoning – Writing proofs
- 5. 5 To do well you should: • Study with pen and paper • Ask for help immediately • Practice, practice, practice… • Follow along in class rather than take notes • Ask questions in class • Keep up with the class • Read the book, not just the slides
- 6. 6 Reasoning about problems • 0.999999999999999….=1? • There exists integers a,b,c that satisfy the equation a2+b2 = c2 • The program below that I wrote works correctly for all possible inputs….. • The program that I wrote never hangs (i.e. always terminates)…
- 7. 7 Tools for reasoning: Logic Ch. 1: Introduction to Propositional Logic • Truth values, truth tables • Boolean logic: • Implications:
- 8. 8 Why study propositional logic? • A formal mathematical “language” for precise reasoning. • Start with propositions. • Add other constructs like negation, conjunction, disjunction, implication etc. • All of these are based on ideas we use daily to reason about things.
- 9. 9 Propositions • Declarative sentence • Must be either True or False. Propositions: • York University is in Toronto • York University is in downtown Toronto • All students at York are Computer Sc. majors. Not propositions: • Do you like this class? • There are x students in this class.
- 10. 10 Propositions - 2 • Truth value: True or False • Variables: p,q,r,s,… • Negation: • p (“not p”) • Truth tables p p T F F T
- 11. 11 Caveat: negating propositions p: “it is not the case that p is true” p: “it rained more than 20 inches in TO” p: “John has many iPads” Practice: Questions 1-7 page 12. Q10 (a) p: “the election is decided”
- 12. 12 Conjunction, Disjunction • Conjunction: p q [“and”] • Disjunction: p q [“or”] p q p q p q T T T T T F F T F T F T F F F F
- 13. 13 Examples • Q11, page 13 p: It is below freezing q: It is snowing (a) It is below freezing and snowing (b) It is below freezing but now snowing (d) It is either snowing or below freezing (or both)
- 14. 14 Exclusive OR (XOR) • p q – T if p and q have different truth values, F otherwise • Colloquially, we often use OR ambiguously – “an entrée comes with soup or salad” implies XOR, but “students can take MATH XXXX if they have taken MATH 2320 or MATH 1019” usually means the normal OR (so a student who has taken both is still eligible for MATH XXXX).
- 15. 15 Conditional • p q [“if p then q”] • p: hypothesis, q: conclusion • E.g.: “If you turn in a homework late, it will not be graded”; “If you get 100% in this course, you will get an A+”. • TRICKY: Is p q TRUE if p is FALSE? YES!! • Think of “If you get 100% in this course, you will get an A+” as a promise – is the promise violated if someone gets 50% and does not receive an A+?
- 16. 16 Conditional - 2 • p q [“if p then q”] • Truth table: p q p q p q T T T T T F F F F T T T F F T T Note the truth table of p q
- 17. 17 Logical Equivalence • p q and p q are logically equivalent • Truth tables are the simplest way to prove such facts. • We will learn other ways later.
- 18. 18 Contrapositive • Contrapositive of p q is q p • Any conditional and its contrapositive are logically equivalent (have the same truth table) – Check by writing down the truth table. • E.g. The contrapositive of “If you get 100% in this course, you will get an A+” is “If you do not get an A+ in this course, you did not get 100%”.
- 19. 19 E.g.: Proof using contrapositive Prove: If x2 is even, x is even • Proof 1: x2 = 2a for some integer a. Since 2 is prime, 2 must divide x. • Proof 2: if x is not even, x is odd. Therefore x2 is odd. This is the contrapositive of the original assertion.
- 20. 20 Converse • Converse of p q is q p • Not logically equivalent to conditional • Ex 1: “If you get 100% in this course, you will get an A+” and “If you get an A+ in this course, you scored 100%” are not equivalent. • Ex 2: If you won the lottery, you are rich.
- 21. 21 Other conditionals Inverse: • inverse of p q is p q • How is this related to the converse? Biconditional: • “If and only if” • True if p,q have same truth values, false otherwise. Q: How is this related to XOR? • Can also be defined as (p q) (q p)
- 22. 22 Example • Q16(c) 1+1=3 if and only if monkeys can fly.
- 23. 23 Readings and notes • Read pages 1-12. • Think about the notion of truth tables. • Master the rationale behind the definition of conditionals. • Practice translating English sentences to propositional logic statements.
- 24. 24 Next Ch. 1.2, 1.3: Propositional Logic - contd – Compound propositions, precedence rules – Tautologies and logical equivalences – Read only the first section called “Translating English Sentences” in 1.2.
- 25. 25 Compound Propositions • Example: p q r : Could be interpreted as (p q) r or p (q r) • precedence order: (IMP!) (Overruled by brackets) • We use this order to compute truth values of compound propositions.
- 26. 26 Tautology • A compound proposition that is always TRUE, e.g. q q • Logical equivalence redefined: p,q are logical equivalences if p q is a tautology. Symbolically p q. • Intuition: p q is true precisely when p,q have the same truth values.
- 27. 27 Manipulating Propositions • Compound propositions can be simplified by using simple rules. • Read page 25 - 28. • Some are obvious, e.g. Identity, Domination, Idempotence, double negation, commutativity, associativity • Less obvious: Distributive, De Morgan’s laws, Absorption
- 28. 28 Distributive Laws p (q r) (p q) (p r) Intuition (not a proof!) – For the LHS to be true: p must be true and q or r must be true. This is the same as saying p and q must be true or p and r must be true. p (q r) (p q) (p r) Intuition (less obvious) – For the LHS to be true: p must be true or both q and r must be true. This is the same as saying p or q must be true and p or r must be true. Proof: use truth tables.
- 29. 29 De Morgan’s Laws (q r) q r Intuition – For the LHS to be true: neither q nor r can be true. This is the same as saying q and r must be false. (q r) q r Intuition – For the LHS to be true: q r must be false. This is the same as saying q or r must be false. Proof: use truth tables.
- 30. 30 Using the laws • Q: Is p (p q) a tautology? • Can use truth tables • Can write a compound proposition and simplify
- 31. 31 Limitations of Propositional Logic • What can we NOT express using predicates? Ex: How do you make a statement about all even integers? If x >2 then x2 >4 • A more general language: Predicate logic (Sec 1.4)
- 32. 32 Next: Predicate Logic Ch 1.4 –Predicates and quantifiers –Rules of Inference
- 33. 33 Predicate Logic • A predicate is a proposition that is a function of one or more variables. E.g.: P(x): x is an even number. So P(1) is false, P(2) is true,…. • Examples of predicates: – Domain ASCII characters - IsAlpha(x) : TRUE iff x is an alphabetical character. – Domain floating point numbers - IsInt(x): TRUE iff x is an integer. – Domain integers: Prime(x) - TRUE if x is prime, FALSE otherwise.
- 34. 34 Quantifiers • describes the values of a variable that make the predicate true. E.g. x P(x) • Domain or universe: range of values of a variable (sometimes implicit)
- 35. 35 Two Popular Quantifiers • Universal: x P(x) – “P(x) for all x in the domain” • Existential: x P(x) – “P(x) for some x in the domain” or “there exists x such that P(x) is TRUE”. • Either is meaningless if the domain is not known/specified. • Examples (domain real numbers) – x (x2 >= 0) – x (x >1) – (x>1) (x2 > x) – quantifier with restricted domain
- 36. 36 Using Quantifiers Domain integers: • Using implications: The cube of all negative integers is negative. x (x < 0) (x3 < 0) • Expressing sums : n n ( i = n(n+1)/2) i=1 Aside: summation notation
- 37. 37 Scope of Quantifiers • have higher precedence than operators from Propositional Logic; so x P(x) Q(x) is not logically equivalent to x (P(x) Q(x)) • x (P(x) Q(x)) x R(x) Say P(x): x is odd, Q(x): x is divisible by 3, R(x): (x=0) (2x >x) • Logical Equivalence: P Q iff they have same truth value no matter which domain is used and no matter which predicates are assigned to predicate variables.
- 38. 38 Negation of Quantifiers • “There is no student who can …” • “Not all professors are bad….” • “There is no Toronto Raptor that can dunk like Vince …” • x P(x) x P(x) why? • x P(x) x P(x) • Careful: The negation of “Every Canadian loves Hockey” is NOT “No Canadian loves Hockey”! Many, many students make this mistake!
- 39. 39 Nested Quantifiers • Allows simultaneous quantification of many variables. • E.g. – domain integers, x y z x2 + y2 = z2 • n x y z xn + yn = zn (Fermat’s Last Theorem) • Domain real numbers: x y z (x < z < y) (y < z < x) Is this true?
- 40. 40 Nested Quantifiers - 2 x y (x + y = 0) is true over the integers • Assume an arbitrary integer x. • To show that there exists a y that satisfies the requirement of the predicate, choose y = -x. Clearly y is an integer, and thus is in the domain. • So x + y = x + (-x) = x – x = 0. • Since we assumed nothing about x (other than it is an integer), the argument holds for any integer x. • Therefore, the predicate is TRUE.
- 41. 41 Nested Quantifiers - 3 • Caveat: In general, order matters! Consider the following propositions over the integer domain: x y (x < y) and y x (x < y) • x y (x < y) : “there is no maximum integer” • y x (x < y) : “there is a maximum integer” • Not the same meaning at all!!!