Announcements. CompSci 230 Discrete Math for Computer Science. Test 1


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1 CompSci 230 Discrete Math for Computer Science Sep 26, 2013 Announcements Exam 1 is Tuesday, Oct. 1 No class, Oct 3, No recitation Oct 47 Prof. Rodger is out Sep 30Oct 4 There is Recitation: Sept Prof. Rodger Slides modified from Rosen Test 1 Closed book, closed notes, closed neighbor There will be a handout of formulas supplied Topics: Reading covers  Chap. 1, 2, and 13.3 Logic, Sets, Functions, Sequences, Cardinality, DFA Chap 2.5 Cardinality How many elements? Can you list them in an ordered way so you don t miss any of them? 20 elements, 5, 10, 15, 100 Infinite, you can list them out in an ordered way: 5, 10, 15, countable set All the subsets of Infinite, you CANNOT list them all out in an ordered way uncountable set
2 Cardinality Definition: The cardinality of a set A is equal to the cardinality of a set B, denoted A = B, if and only if there is a onetoone correspondence (i.e., a bijection) from A to B. If there is a onetoone function (i.e., an injection) from A to B, the cardinality of A is less than or the same as the cardinality of B and we write A B A B A B A B A B Cardinality Definition: A set that is either finite or has the same cardinality as the set of positive integers (Z + ) is called countable. A set that is not countable is uncountable. All subsets of set of real numbers R countable uncountable uncountable When an infinite set is countable (countably infinite) its cardinality is. We write S = S Showing that a Set is Countable An infinite set is countable if and only if it is possible to list the elements of the set in a sequence (indexed by the positive integers). The reason for this is that a onetoone correspondence f from the set of positive integers to a set S can be expressed in terms of a sequence a 1,a 2,, a n, where a 1 = f( ), a 2 = f( ),, a n = f(n), Hilbert s Grand Hotel David Hilbert The Grand Hotel (example due to David Hilbert) has countably infinite number of rooms, each occupied by a guest. We can always accommodate a new guest at this hotel. How is this possible? Explanation: Because the rooms of Grand Hotel are countable, we can list them as Room, Room, Room, and so on. When a new guest arrives, we move the guest in Room to Room, the guest in Room to Room, and in general the guest in Room n to Room n +, for all positive integers n. This frees up Room, which we assign to the new guest, and all the current guests still have rooms. The hotel can also accommodate a countable number of new guests, all the guests on a countable number of buses where each bus contains a countable number of guests
3 Showing that a Set is Countable Example : Show that the set of positive even integers E is a countable set. Solution: Let f x x. Then f is a bijection from N to E since f is both onetoone and onto. To show that it is onetoone, suppose that f f Then, and so. To see that it is onto, suppose that t is an even positive integer. Then t = k for some positive integer k and f(k) = t. Showing that a Set is Countable Example : Show that the set of integers Z is countable. Solution: Can list in a sequence: Or can define a bijection from N to Z: When n is even: f(n) = n/ When n is odd: f(n) = (n )/ The Positive Rational Numbers are Countable Definition: A rational number can be expressed as the ratio of two integers p and q such that q ¾ is a rational number is not a rational number. Example : Show that the positive rational numbers are countable. Solution: The positive rational numbers are countable since they can be arranged in a sequence: r 1, r 2, r 3, For p/q Does this work? List all those numbers with q=1 1/1, 2/1, 3/1, 4/1, 5/1, Then list all those numbers with q=2 1/2, 2/2, 2/3, 2/4, 2/5, Then list all those numbers with q=3, etc Doesn t work, infinite in too many directions!
4 Solution for listing out all p/q List all s.t. p+q=2 Then all s.t. p+q=3 Then all s.t. p+q=4 Etc Why does this work? Example  Strings Show that the set S of strings over the alphabet {0,1} is countable. What is S? Is S infinite? S={010, 1101, , } Solution: List out all the strings of length 0, then 1, then 2, etc. There are a finite number of each S={ Example: Is the set of all Java programs countable? Yes Solution: } is finite S is the set of all strings over the alphabet J = is a valid Java program} List out all the strings in S and if they compile (then a valid Java program), put them in J The Real numbers are uncountable Proof(sketch) by diagonalization Suppose we can list out all the real numbers without missing any of them.
5 List out all real numbers Claim we missed one! There is a number whose ith digit is different from the ith digit in the ith number. Contradiction! Thus real numbers are not countable a Matrices Examples: Graph Theory  Express which vertices of a graph are connected by edges Graphics Represent a 3D object with a matrix Project a 3D object onto a 2D screen Optimal curve fitting Transportation systems. Communication networks Economics and Game Theory For now, definition and basic operations Matrix Definition: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an matrix. The plural of matrix is matrices. A matrix with the same number of rows as columns is called square. Two matrices are equal if they have the same number of rows and the same number of columns and the corresponding entries in every position are equal. Notation Let m and n be positive integers and let The ith row of A is the a i1, a i2,,a in j A m The (i,j)th element or entry of A is the element a ij. We can use A = [a ij ] to denote the matrix with its (i,j)th element equal to a ij.
6 Matrix Arithmetic: Addition Defintion: Let and be matrices. The sum of A and B, denoted by A + B, is the a b A + B = [a ij b ij Example Matrix Multiplication Definition: Let A be an matrix and B be a. The product of A and B, denoted by AB, is the i,j i j AB = [c ij c ij a i b a i b j a kj b j Example Illustration of Matrix Multiplication The Product of A = [ ij B = [ ij Example use of Matrix Multiplication Solving set of linear equations
7 Is Matrix Multiplication Commutative Example: Let Does AB = BA? Solution: AB BA Identity Matrix and Powers of Matrices Definition: The identity matrix of order n is the m n matrix I n = [ ij ], where ij i j ij i j AI n = I m A A m n n n matrix, we have: A I n A r AAA A r times Transposes of Matrices Definition: Let A = [a ij m n A A,is the n m matrix obtained by interchanging the rows and columns of A. If A =[b ij ], then i,n j... m
Cardinality. The set of all finite strings over the alphabet of lowercase letters is countable. The set of real numbers R is an uncountable set.
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