Lecture 11 Countable and uncountable sets. Matrices. Instructor: Kangil Kim (CSE) E-mail: kikim01@konkuk.ac.kr Tel. : 02-450-3493 Room : New Milenium Bldg. 1103 Lab : New Engineering Bldg. 1202 Next topic: Course syllabus Logic and proofs Sets Functions Integers and modular arithmetic Sequences and summations Counting Probability Relations Graphs 1
What We Have Leanrt Sequence and Summation - Simple Sequences - Recurrent Relation - More Sequences - Summation of Elements in Sequences Counting - Countability - Countable Sets - Properties of Countability What We Will Learn Matrices - Basic Concept - Basic Matrices
Arithmetic series Definition: The sum of the terms of the arithmetic progression a, a+d,a+2d,, a+nd is called an arithmetic series. Theorem: The sum of the terms of the arithmetic progression a, a+d,a+2d,, a+nd is n S ( a jd) na d j 1 j 1 n (n 1) j na d n 2
Geometric series Definition: The sum of the terms of a geometric progression a, ar, ar 2,..., ar k is called a geometric series. Theorem: The sum of the terms of a geometric progression a, ar, ar 2,..., ar n is n n n 1 j j r 1 S ( ar ) a r a j 0 j 0 r 1 Infinite geometric series Infinite geometric series can be computed in the closed form for x<1 How? k k 1 n n x 1 1 1 x lim k x lim k x 1 x 1 1 x n 0 n 0 Thus: n 1 x 1 n 0 x 2
Cardinality Recall: The cardinality of a finite set is defined by the number of elements in the set. Definition: The sets A and B have the same cardinality if there is a one-to-one correspondence between elements in A and B. In other words if there is a bijection from A to B. Recall bijection is one-to-one and onto. Assume A = {a,b,c} and B = {α,β,γ} and function f defined as: a α b β c γ If there is a one-toone function from A to B and from B to A, A and B are one-to-one correspondence. F defines a bijection. Therefore A and B have the same cardinality, i.e. A = B = 3. 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 called uncountable. Why these are called countable? The elements of the set can be enumerated and listed. 3
Countable sets Assume A = {0, 2, 4, 6,... } set of even numbers. Is it countable? Countable sets Assume A = {0, 2, 4, 6,... } set of even numbers. Is it countable? Using the definition: Is there a bijective function f: Z + A Z+ = {1, 2, 3, 4, } 4
Countable sets Assume A = {0, 2, 4, 6,... } set of even numbers. Is it countable? Using the definition: Is there a bijective function f: Z + A Z+ = {1, 2, 3, 4, } Define a function f: x 2x - 2 (an arithmetic progression) 1 2(1)-2 = 0 2 2(2)-2 = 2 3 2(3)-2 = 4... Countable sets Assume A = {0, 2, 4, 6,... } set of even numbers. Is it countable? Using the definition: Is there a bijective function f: Z + A Z+ = {1, 2, 3, 4, } Define a function f: x 2x - 2 (an arithmetic progression) 1 2(1)-2 = 0 2 2(2)-2 = 2 3 2(3)-2 = 4... one-to-one (why?) 5
Countable sets Assume A = {0, 2, 4, 6,... } set of even numbers. Is it countable? Using the definition: Is there a bijective function f: Z + A Z+ = {1, 2, 3, 4, } Define a function f: x 2x - 2 (an arithmetic progression) 1 2(1)-2 = 0 2 2(2)-2 = 2 3 2(3)-2 = 4... one-to-one (why?) 2x-2 = 2y-2 => 2x = 2y =>x = y. onto (why?) a A, (a+2) / 2 is the pre-image in Z +. Therefore A = Z +. Countable sets Theorem: The set of integers Z is countable. Solution: Can list a sequence: 0, 1, 1, 2, 2, 3, 3,.. Or can define a bijection from Z + to Z: When n is even: f(n) = n/2 When n is odd: f(n) = (n 1)/2 6
Countable sets Definition: A rational number can be expressed as the ratio of two integers p and q such that q 0. ¾ is a rational number 2is not a rational number. Theorem: 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, Countable sets Theorem: The positive rational numbers are countable. First row q = 1. Second row q = 2. etc. Constructing the List First list p/q with p + q = 2. Next list p/q with p + q = 3 And so on. 7
Cardinality Theorem: The set of real numbers (R) is an uncountable set. Proof by a contradiction. 1) Assume that the real numbers are countable. 2) Then every subset of the reals is countable, in particular, the interval from 0 to 1 is countable. This implies the elements of this set can be listed say r1, r2, r3,... where r1 = 0.d 11 d 12 d 13 d 14... r2 = 0.d 21 d 22 d 23 d 24... r3 = 0.d 31 d 32 d 33 d 34...... where the d ij {0,1,2,3,4,5,6,7,8,9}. Real numbers are uncountable Proof cont. 3) Want to show that not all reals in the interval between 0 and 1 are in this list. Form a new number called r = 0.d 1 d 2 d 3 d 4... where d i = 2, if d ii 2 3 if d ii = 2 suppose r1 = 0.75243... d1 = 2 r2 = 0.524310... d2 = 3 r3 = 0.131257... d3 = 2 r4 = 0.9363633... d4 = 2...... rt = 0.23222222... dt = 3 8
Real numbers are uncountable r = 0.d 1 d 2 d 3 d 4... where 2, if d ii 2 d i = 3 if d ii = 2 Claim: r is different than each member in the list. Is each expansion unique? Yes, if we exclude an infinite string of 9s..02850 =.02849 Therefore r and r i differ in the i-th decimal place for all i. -> Contradiction, we can not list all real numbers in the countable way. Useful Countability Property If A and B are countable sets, then A B is also countable i) finite and finite ii) countably infinite and finite iii) countably infinite and coutnably infinite i) the union is finite -> countable ii) concatenation of the finite and countably infinite is countable iii) A set locating elements at the same index of the two sets is countable 9
Useful Countability Property Any set with an uncountable subset is uncountable Any subset of a countable set is countable -> The restriction of an injective function to a subset of its domain is still injective. If S is a countable set and x S, then S {x} is countable. ->Let f: S N be an injection. Define g: S {x} N by g(x) = 0 and g(y) = f(y) + 1 for all y in S. This function g is an injection. Matrices
Matrices Definitions: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n matrix. Note: The plural of matrix is matrices. Matrices Definitions: A matrix is a rectangular array of numbers. A matrix with m rows and n columns is called an m x n matrix. Note: The plural of matrix is matrices. Definitions: A matrix with the same number of rows as columns is called a square matrix. 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. 10
Matrices Let m and n be positive integers and let The ith row of A is the 1 x n matrix [a i1, a i2,,a in ]. The jth column of A is the m x 1matrix: 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. Matrix addition Defintion: Let A a ij and B b ij be m x n matrices. The sum of A and B, denoted by A + B, is the m x n matrix that has a ij b ij as its i,j th element. In other words, A + B = [a ij b ij. Note: matrices of different sizes can not be added. 11
Matrix multiplication Definition: Let A be an m x k matrix and B be a k x n matrix. The product of A and B, denoted by AB, is the m x n matrix that has its i,j th element equal to the sum of the products of the corresponding elments from the ith row of A and the jth column of B. In other words, if AB = [c ij then c ij a i1 b 1j a i2 b 2j a jk b kj. The product is not defined when the number of columns in the first matrix is not equal to the number of rows in the second matrix Matrix multiplication The Product of A = [a ij and B = [b ij 12
Matrix multiplication 2 4 3 1 1 2 2 4 6 * 1 4 2 1 2 2 1 5 3 =????????? Matrix multiplication Properties of matrix multiplication: Does AB = BA? AB BA 13
Matrix multiplication Properties of matrix multiplication: Does AB = BA? AB:? Matrix multiplication Properties of matrix multiplication: Does AB = BA? AB: 3 14
Matrix multiplication Properties of matrix multiplication: Does AB = BA? AB: BA:? 3 Matrix multiplication Properties of matrix multiplication: Does AB = BA? AB: BA: 3 Conclusion: AB BA 15
Matrices Definition: The identity matrix (of order n) is the n x n matrix I n = [ ij ], where ij 1 if i j and ij 0 if i j. Properties: Assume A is an m x n matrix. Then: AI n A and I m A A Assume A is an n x n matrix. Then: A 0 I n Matrices Definition: Powers of square matrices When A is an n n matrix, we have: A 0 I n A r AAA A r 16
Matrix transpose Definition: Let A = [a ij ] be an m x n matrix. The transpose of A, denoted by A T,is the n x m matrix obtained by interchanging the rows and columns of A. If A T =[b ij ], then b ij a ji for i 1,2,,n and j 1,2,...,m. Matrix inverse Definition: Let A = [a ij ] be an n x n matrix. The inverse of A, denoted by A -1, is the n x m matrix such that A A -1 = A -1 A = I Note: the inverse of the matrix A may not exist. 17