Sets. Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, Outline Sets Equality Subset Empty Set Cardinality Power Set

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1 Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 6, 2012 Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

2 Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

3 What is a Set? A set is an unordered set of collection The objects in a set are called elements or members of the set a A : a is an element of A, a belongs to A a / A : a is not an element of A, a does not belong to A One way to describe a set is to list all its elements {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} the set of digits {a, b, c, d,..., x, y, z} the set of Latin letters, alphabet A set can be an element of another set {{a,..., x, y, z}, {α, β, γ,...},...} the set of alphabets Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

4 Set Builder Large sets can be described using set builder Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

5 Set Builder Large sets can be described using set builder {x p(x)} the set of all x such that p(x) Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

6 Set Builder Large sets can be described using set builder {x p(x)} the set of all x such that p(x) {x there is y such that x = 2y} the set of all even numbers Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

7 Set Builder Large sets can be described using set builder {x p(x)} the set of all x such that p(x) {x there is y such that x = 2y} the set of all even numbers {x y (x = 2y)} the set of all even numbers Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

8 Set Builder Large sets can be described using set builder {x p(x)} the set of all x such that p(x) {x there is y such that x = 2y} the set of all even numbers {x y (x = 2y)} the set of all even numbers {x x is a black cow} the set of all black cows Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

9 Some Useful N = {0, 1, 2, 3,...} : the set of natural numbers Z = {..., 3, 2, 1, 0, 1, 2, 3,...} : the set of integers Q = { p q p, q are integers and q 0} : the set of rational numbers Z + : the set of positive integers Q + : the set of positive rationals R : the set of real numbers Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

10 Russel s Paradox We should use the set builder construction very carefully Our understanding of sets is very broad. For example, a set can be its own element Let U be the set of all sets that do not contain itself as an element: U = {x x / x} Does U contain itself? Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

11 Russel s Paradox We should use the set builder construction very carefully Our understanding of sets is very broad. For example, a set can be its own element Let U be the set of all sets that do not contain itself as an element: U = {x x / x} Does U contain itself? If yes, then U U, hence U does not satisfy the condition, and therefore U / U Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

12 Russel s Paradox We should use the set builder construction very carefully Our understanding of sets is very broad. For example, a set can be its own element Let U be the set of all sets that do not contain itself as an element: U = {x x / x} Does U contain itself? If yes, then U U, hence U does not satisfy the condition, and therefore U / U If not, then U / U, hence U does satisfy the condition, and therefore U U Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

13 Universe In theory the elements of sets can be anything. In practice, however, it is not such a good idea to allow such diversity This is why we usually have some sort of universal set or universe in mind, that contains all objects we need Example {x 1 x 10} : what is this set? {x Z 1 x 10} : set of all integers between 1 and 10 {x Q 1 x 10} : set of all rationals between 1 and 10 Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

14 Two sets are equal if and only if they have the same elements That is A = B if and only if x (x A x B) Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

15 Two sets are equal if and only if they have the same elements That is A = B if and only if x (x A x B) {1, 3, 5} = {3, 1, 5} Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

16 Two sets are equal if and only if they have the same elements That is A = B if and only if x (x A x B) {1, 3, 5} = {3, 1, 5} {1, 3, 5} = {3, 1, 1, 5, 5, 5, 3, 1, 1, 5} Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

17 Two sets are equal if and only if they have the same elements That is A = B if and only if x (x A x B) {1, 3, 5} = {3, 1, 5} {1, 3, 5} = {3, 1, 1, 5, 5, 5, 3, 1, 1, 5} {1, 3, 5} {3, {1}, 5} Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

18 Set B is a subset of set A if every element of B is also an element of A That is B A if and only if x (x B x A) Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

19 Set B is a subset of set A if every element of B is also an element of A That is B A if and only if x (x B x A) {1, 5} {3, 1, 5} Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

20 Set B is a subset of set A if every element of B is also an element of A That is B A if and only if x (x B x A) {1, 5} {3, 1, 5} {1, 3, 5} {3, 1, 1, 5, 5, 5, 3, 1, 1, 5} Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

21 Set B is a subset of set A if every element of B is also an element of A That is B A if and only if x (x B x A) {1, 5} {3, 1, 5} {1, 3, 5} {3, 1, 1, 5, 5, 5, 3, 1, 1, 5} {1} {3, {1}, 5} Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

22 Set B is a subset of set A if every element of B is also an element of A That is B A if and only if x (x B x A) {1, 5} {3, 1, 5} {1, 3, 5} {3, 1, 1, 5, 5, 5, 3, 1, 1, 5} {1} {3, {1}, 5} Set B is not a subset of set A if x (x B x / A) There is an element in B that is not an element of A Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

23 Set B is a subset of set A if every element of B is also an element of A That is B A if and only if x (x B x A) {1, 5} {3, 1, 5} {1, 3, 5} {3, 1, 1, 5, 5, 5, 3, 1, 1, 5} {1} {3, {1}, 5} Set B is not a subset of set A if x (x B x / A) There is an element in B that is not an element of A Another way of saying that two sets are equal is that each of them is a subset of the other A = B if and only if (A B) (B A) Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

24 Proper Set B is a proper subset of set A if it is a subset of A and is not equal to A That is B A if and only if (B A) ( x (x A x / B)) Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

25 Syllogism Theorem If A B and B C, then A C Proof. We have to prove that, for every x, if x A, then x C. Take any x such that x A. Since A B, this implies x B. Next, as B C and x B, we have x C. Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

26 Empty set is the set that has no elements and denoted by Theorem For any set A, (i) A, and (ii) A A Proof. (i) We have to prove that x (x x A). For any c, c is FALSE, hence c c A is TRUE. Then we use the rule of universal generalization. (ii) homework! Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

27 Let A be a set. If there are exactly n distinct elements in A where n is a natural number, then we say that A is a finite set and that n is the cardinality of A denoted by A = n Examples = 0 {1, 1, 2, 3, 3, 3} = 3 {1, {2, 3}, 4, 5} = 4 that are not finite are said to be infinite. Examples of infinite sets include R, Q, Z, N Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

28 Power set Given a set A, the power set of A is the set of all subsets of A denoted by P(A) Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

29 Power set Given a set A, the power set of A is the set of all subsets of A denoted by P(A) Example A = {1, 2, A} Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

30 Power set Given a set A, the power set of A is the set of all subsets of A denoted by P(A) Example A = {1, 2, A} P(A) = {, {1}, {2}, {A}, {1, 2}, {1, A}, {2, A}, {1, 2, A}} Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

31 Power set Theorem If A is a finite set, then P(A) = 2 A Proof. Let A = {a 1, a 2, a 3,..., a n }. Represent each subset S of A as a string s of n bits (0 1) a i S s[i] = 1 [1]. For instance, is encoded as A itself as Such a representation is called Grey code in Grimaldi and characteristic function elsewhere Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

32 Power set Proof cont. We observe that for each bit there are two possibilities, and its value does not depend on the values of other bits. Thus we have 2x2x2x... x2 = 2 A possible subsets Gazihan Alankuş (Based on original slides by Brahim Hnich et al.)

Cardinality and Bijections

Countable and Cardinality and Bijections Gazihan Alankuş (Based on original slides by Brahim Hnich et al.) August 13, 2012 Countable and Countable and Countable and How to count elements in a set? How