SFWR ENG 2FA3: Discrete Mathematics and Logic II

Transcription

1 Mathematics and Outline Dr. Ridha Khedri 1 1 Department of Computing and Software, McMaster University Canada L8S 4L7, Hamilton, Ontario

2 Outline of Part I 1 Main topics in the course outline Topics Outline Part I: Course Outline Presentation Part II: Remarks

3 Outline of Part II 2 3 Outline Part I: Course Outline Presentation Part II: Remarks 4 5

4 Part I Course Outline Course Outline Presentation

5 Main topics in the course outline Course objective Textbooks Course Information on Web Assignments and exams Policy Statements and Notes Detailed course outline References Discrimination Academic Integrity Course Outline Topics

6 Part II Today s Lecture

7 Why software Engineers need? Formal specification of systems Formal verification of systems Understand the various models of computation and their limitations...

8 Shut off the pumps if the water level is above 100 meters for 4 seconds. There are several reasonable interpretations for this sentence. 1 Shut off the pumps if the mean water level over the past 4 seconds was above 100 meters. [ ( T T 4 WL(t)dt ) ] 4 > 100

9 2 Shut off the pumps if the median water level over the past 4 seconds was above 100 meters. (Max [t 4,t] (WL(t)) + Min [t 4,t] (WL(t))) 2 > Shut off the pumps if the minimum water level over the past 4 seconds was above 100 meters. Min [t 4,t] (WL(t)) >

10 Television channels are assigned to broadcasting stations by a governmental agency. Obviously, two stations in geographic proximity must get different channels, to avoid reception interference. Suppose that the rule has been adopted that stations within 140 miles of each other (as the crow flies) must have different channels. The grid shows the locations of 15 hypothetical stations. Each square is 50 miles on a side. How many channels are required, and how can they be assigned to comply with the rule? B C E N D F G H J L A I K M O

11 C E N B D F G H J L A I M K O

12 (finite ) is the study of mathematical structures that are fundamentally discrete The objects studied are countable sets, such as integers, finite graphs, and formal languages has become popular in recent decades because of its applications to computer science Concepts and s from discrete are useful to study or describe objects or problems in computer algorithms and programming languages

13 includes the following topics: Logic - a study of reasoning; Set theory - a study of collections of elements; Number theory - a study of the properties of numbers in general, and integers in particular; Combinatorics - a study of counting; Graph theory - a study of graphs and the algorithms on graphs; Algorithmics - a study of methods of calculation; Computability and complexity theories - dealing with theoretical and practical limitations of algorithms; Algebras etc.

14 Continuous is the study of mathematical structures that are fundamentally continuous The objects studied are uncountable sets, such as reals Continuous includes the following topics: Analysis - Limits, continuity and differentiability, Taylor series, Complex numbers, etc. Fourier series Representation of signals - Fourier transforms, Laplace transforms, etc.

15 No can be made easy, and discrete is no exception read with pencil in hand and a pad of paper beside you Try to solve the problem before reading its solution you may be able to solve it = you understand the concepts After working on it, you are not able to solve it = you put your finger on what you do not understand You need to pay careful attention to being precise You should make up your own dictionary as you study DO THE EXERCISES Justify your answers

16 Conventional Proof of A (B C) = (A B) (A C) We first show that A (B C) (A B) (A C). If x A (B C), then either x A or x B C. If x A, then certainly x A B and x A C, so x (A B) (A C). On the other hand, if x B C, then x B and x C, so x A B and x A C, so x (A B) (A C). Hence, A (B C) (A B) (A C). Conversely, if y (A B) (A C), then y A B and y A C. We consider two cases: y A and y A. If y A, then y A (B C), and this part is done. If y A, then, since y A B we must have y B. Similarly, since y A C and y A, we have y C. Thus, y B C, and this implies y A (B C). Hence(A B) (A C) A (B C).

17 Calculational Proof of A (B C) = (A B) (A C) x A (B C) Definition of x A x B C Definition of x A x B x C Distributivity of over (x A x B) (x A x C) Definition of, twice x (A B) x (A C) Definition of x (A B) (A C)

18 The last presentation of the proof is obvious and straightforward Anyone with a little experience in such calculational proofs will have no difficulty reproducing them These proofs are rigorous and could be checked by a mechanical proof checker

19 expression 0 op 0 hint 0 expression 1 op 1 hint 1 expression 2 expression n where op i, i = 0,, n 1, is a relational operator =, <, >,,,,, like it could be a logical operator

20 Prove that 2 (5 x 2 2 x + 6 x + X 2 4) = 12 X 2 + 8(X 1) 2 (5 x 2 2 x + 6 x + X 2 4) = Distributivity of * over + & calculus 10 x 2 4 x + 12 x + 2 X 2 8 = Commutativity of + & calculus 12 X x 8 = Distributivity of * over + 12 X (x 1)

21 more examples 3x + 3 = 0 3x = 3 Add 3 in the two sides of the equation & 3 3 = 0 & 0 is the neutral element for & divide by 3 the two sides of the equation & 3 3 = 1 x = 1 When appropriate, we adopt this calculational way to present proofs.

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