Discrete Structures - CM0246 Cardinality

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1 Discrete Structures - CM0246 Cardinality Andrés Sicard-Ramírez Universidad EAFIT Semester

2 Cardinality Definition (Cardinality (finite sets)) Let A be a set. The number of (distinct) elements in A, denoted A, is called the cardinality of A. Cardinality 2/26

3 Cardinality Definition (Cardinality (finite sets)) Let A be a set. The number of (distinct) elements in A, denoted A, is called the cardinality of A. Definition (Cardinality (finite and infinite sets)) The sets A and B have the same cardinality, if and only, there is a bijection from A to B. Cardinality 3/26

4 Cardinality Definition (Cardinality (finite sets)) Let A be a set. The number of (distinct) elements in A, denoted A, is called the cardinality of A. Definition (Cardinality (finite and infinite sets)) The sets A and B have the same cardinality, if and only, there is a bijection from A to B. Injunction, surjection or bijection? Draw figures in the whiteboard. Cardinality 4/26

5 Cardinality Examples Z + = N Cardinality 5/26

6 Cardinality Examples Z + = N N = Even, where Even is the set defined by Even = {2n n N}. Cardinality 6/26

7 Cardinality Examples Z + = N N = Even, where Even is the set defined by Even = {2n n N}. N = Even, where M k is the set of the non-negative multiples of k Z +, i.e. M k = {nk n N}. Cardinality 7/26

8 Cardinality Examples Z + = N N = Even, where Even is the set defined by Even = {2n n N}. N = Even, where M k is the set of the non-negative multiples of k Z +, i.e. M k = {nk n N}. [0, 1] = [a, b], where a, b R and a < b. Cardinality 8/26

9 Cardinality The possibility that whole and part may have the same number of terms is, it must be confessed, shocking to common-sense. (Russell 1903, p. 358) ( ) Cardinality 9/26

10 Cardinality Example (Lipschutz (1998), Solved problem 6.2, p. 153) Prove that [0, 1] = (0, 1). Note that [0, 1] = {0, 1, 1/2, 1/3, 1/4, } A (0, 1) = {1/2, 1/3, 1/4, } A where A = [0, 1] {0, 1, 1/2, 1/3, 1/4, } = (0, 1) {1/2, 1/3, 1/4, }. Cardinality 10/26

11 Cardinality Example (cont.) From the figure we define the bijective function f [0, 1] (0, 1) by 1/2 if x = 0, { f(x) = 1/(n + 1) if x = 1/n where n Z +, { x otherwise. Fig. 6.5 of (Lipschutz 1998). Cardinality 11/26

12 Cardinality Exercise Let A and B be sets. Show A B = B A. Cardinality 12/26

13 Enumerable and Non-Enumerable Sets Has all the infinite sets the same cardinality? Cardinality 13/26

14 Enumerable and Non-Enumerable Sets Has all the infinite sets the same cardinality? Definition (Enumerable set) A set that is either finite or has the same cardinality as the set of positive integers is called enumerable (or countable). Cardinality 14/26

15 Enumerable and Non-Enumerable Sets Has all the infinite sets the same cardinality? Definition (Enumerable set) A set that is either finite or has the same cardinality as the set of positive integers is called enumerable (or countable). Definition (Non-enumerable set) A set that is not enumerable countable is called non-enumerable (or uncountable). Cardinality 15/26

16 Enumerable and Non-Enumerable Sets Has all the infinite sets the same cardinality? Definition (Enumerable set) A set that is either finite or has the same cardinality as the set of positive integers is called enumerable (or countable). Definition (Non-enumerable set) A set that is not enumerable countable is called non-enumerable (or uncountable). Examples Whiteboard Cardinality 16/26

17 Enumerable and Non-Enumerable Sets Example (The positive rational numbers are enumerable ) 2.5 Cardinality of Terms not circled are not listed because they repeat previously listed terms FIGURE 3 The Positive Rational Numbers Are Countable. Remark: We don t define explicitly the function, but a method (program) arrange the positive rational numbers by listing those with denominator q = 1 in the for enumerating thethose set. with denominator q = 2 in the second row, and so on, as displayed in Figure 3. The key to listing the rational numbers in a sequence is to first list the positiv numbers p/q with p + q = 2, followed by those with p + q = 3, followed by th p + q = 4, and so on, following the path shown in Figure 3. Whenever we encounter Fig. 3 of (Rosen 2012, p/q that 2.5). is already listed, we do not list it again. For example, when we come to 2/ Cardinality do not list it because we have already listed 1/1 = 1. The initial terms in the17/26 list o

18 Enumerable and Non-Enumerable Sets Theorem The interval (0, 1) is non-enumerable. Proof (next slide). Cardinality 18/26

19 Enumerable and Non-Enumerable Sets Proof. Let s suppose (0, 1) is enumerable. r 1 = 0.d 11 d 12 d 13 d 14 r 2 = 0.d 21 d 22 d 23 d 24 r 3 = 0.d 31 d 32 d 33 d 34 Cardinality 19/26

20 Enumerable and Non-Enumerable Sets Proof. Let s suppose (0, 1) is enumerable. Let x = 0.d 1 d 2 d 3 (0, 1), where r 1 = 0.d 11 d 12 d 13 d 14 r 2 = 0.d 21 d 22 d 23 d 24 r 3 = 0.d 31 d 32 d 33 d 34 d i = { 4 iff d ii 4, 5 iff d ii = 4. Cardinality 20/26

21 Enumerable and Non-Enumerable Sets Proof. Let s suppose (0, 1) is enumerable. Let x = 0.d 1 d 2 d 3 (0, 1), where r 1 = 0.d 11 d 12 d 13 d 14 r 2 = 0.d 21 d 22 d 23 d 24 r 3 = 0.d 31 d 32 d 33 d 34 d i = { 4 iff d ii 4, 5 iff d ii = 4. The number x doesn t belong to the above enumeration. Therefore (0, 1) is non-enumerable. Cardinality 21/26

22 Enumerable and Non-Enumerable Sets Theorem Let A and B be sets such A B. If A is non-enumerable then B is non-enumerable. Cardinality 22/26

23 Enumerable and Non-Enumerable Sets Theorem The set of the real numbers is non-enumerable. Cardinality 23/26

24 Enumerable and Non-Enumerable Sets Theorem The set of the real numbers is non-enumerable. Proof. The interval (0, 1) is a non-enumerable subset of R. previous theorem), R is non-enumerable. Therefore (using a Cardinality 24/26

25 Enumerable and Non-Enumerable Sets Theorem The set of the real numbers is non-enumerable. Proof. The interval (0, 1) is a non-enumerable subset of R. previous theorem), R is non-enumerable. Therefore (using a Comment about the continuum hypothesis Cardinality 25/26

26 References Lipschutz, S. (1998). Schaum s Outline of Theory and Problems of Set Theory and Related Topics. 2nd ed. McGraw-Hill (cit. on pp. 10, 11). Rosen, K. H. (2012). Discrete Mathematics and Its Applications. 7th ed. McGraw-Hill (cit. on p. 17). Russell, B. (1903). The Principles of Mathematics. W. W. Norton & Company, Inc (cit. on p. 9). Cardinality 26/26

Discrete Structures - CM0246 Mathematical Induction

Discrete Structures - CM0246 Mathematical Induction Andrés Sicard-Ramírez EAFIT University Semester 2014-2 Motivation Example Conjecture a formula for the sum of the first n positive odd integers. Discrete