Presentation is loading. Please wait.

Presentation is loading. Please wait.

Chapter 6: Set Theory 6.1 Set Theory: Definitions and the Element Method of Proof 1 6.1 Set Theory - Definitions and the Element Method of Proof The introduction.

Similar presentations


Presentation on theme: "Chapter 6: Set Theory 6.1 Set Theory: Definitions and the Element Method of Proof 1 6.1 Set Theory - Definitions and the Element Method of Proof The introduction."— Presentation transcript:

1 Chapter 6: Set Theory 6.1 Set Theory: Definitions and the Element Method of Proof Set Theory - Definitions and the Element Method of Proof The introduction of suitable abstractions is our only mental aid to organize and master complexity. – E. W. Dijkstra, 1930 – 2002 Erickson

2 Let’s write what it means for a set A to be a subset of a set B as a formal universal conditional statement: 6.1 Set Theory - Definitions and the Element Method of Proof 2 A  B   x, if x  A then x  B. Erickson

3 The negation is existential 6.1 Set Theory - Definitions and the Element Method of Proof 3 A  B   x, if x  A and x  B. Erickson

4 A proper subset of a set is a subset that is not equal to its containing set. 6.1 Set Theory - Definitions and the Element Method of Proof 4 A is a proper subset of B  1. A  B, and 2. there is at least one element in B that is not in A. Erickson

5 Let sets X and Y be given. To prove that X  Y, 1.Suppose that x is a particular but arbitrarily chosen element of X, 2.Show that x is an element of Y. 6.1 Set Theory - Definitions and the Element Method of Proof 5Erickson

6 Let A = {n   | n = 5r for some integer r} and B = {m   | m = 20s for some integer s}. a.Is A  B? Explain. b.Is B  A? Explain. 6.1 Set Theory - Definitions and the Element Method of Proof 6Erickson

7 Given sets A and B, A equals B, written A = B, iff every element of A is in B and every element of B is in A. Symbolically, 6.1 Set Theory - Definitions and the Element Method of Proof 7 A = B  A  B and B  A Erickson

8 Let A and B be subsets of a universal set U. 6.1 Set Theory - Definitions and the Element Method of Proof 8 1. The union of A and B denoted A  B, is the set of all elements that are in at least one of A or B. Symbolically: A  B = {x  U | x  A or x  B} Erickson

9 Let A and B be subsets of a universal set U. 6.1 Set Theory - Definitions and the Element Method of Proof 9 2. The intersection of A and B denoted A  B, is the set of all elements that are common to both A or B. Symbolically: A  B = {x  U | x  A and x  B} Erickson

10 Let A and B be subsets of a universal set U. 6.1 Set Theory - Definitions and the Element Method of Proof The difference of B minus A (or relative complement of A in B) denoted B – A, is the set of all elements that are in B but not A. Symbolically: B – A = {x  U | x  B and x  A} Erickson

11 Let A and B be subsets of a universal set U. 6.1 Set Theory - Definitions and the Element Method of Proof The complement of A denoted A c, is the set of all elements in U that are not A. Symbolically: A c = {x  U | x  A} Erickson

12 Let the universal set be the set R of all real numbers and let A = {x  R | 0 < x  2}, B = {x  R | 1  x < 4}, and C = {x  R | 3  x < 9}. Find each of the following: a. A  Bb. A  Bc. A c d. A  C e. A  Cf. B c g. A c  B c h. A c  B c i. (A  B) c j. (A  B) c 6.1 Set Theory - Definitions and the Element Method of Proof 12Erickson

13 Given sets A 0, A 1, A 2, … that are subsets of a universal set U and given a nonnegative integer n, 6.1 Set Theory - Definitions and the Element Method of Proof 13Erickson

14 Empty Set A set with no elements is called the empty set (or null set) and denoted by the symbol . Disjoint Two sets are called disjoint iff they have no elements in common. Symbolically: A and B are disjoint  A  B =  6.1 Set Theory - Definitions and the Element Method of Proof 14Erickson

15 Mutually Disjoint Sets A 1, A 2, A 3, … are mutually disjoint (or pairwise disjoint or nonoverlapping) iff no two sets A i and A j with distinct subscripts have any elements in common. More precisely, for all i, j = 1, 2, 3, … A i  A j =  whenever i  j. 6.1 Set Theory - Definitions and the Element Method of Proof 15Erickson

16 Let for all positive integers i. 6.1 Set Theory - Definitions and the Element Method of Proof 16Erickson

17 Partition A finite or infinite collection of nonempty sets {A 1, A 2, A 3, …} is a partition of a set A iff, 1. A is the union of all the A i 2.The sets A 1, A 2, A 3, …are mutually disjoint. 6.1 Set Theory - Definitions and the Element Method of Proof 17Erickson

18 6.1 Set Theory - Definitions and the Element Method of Proof 18Erickson

19 Power Set Given a set A, the power set of A is denoted  (A), is the set of all subsets of A. 6.1 Set Theory - Definitions and the Element Method of Proof 19Erickson

20 Suppose A = {1, 2} and B = {2, 3}. Find each of the following: 6.1 Set Theory - Definitions and the Element Method of Proof 20Erickson


Download ppt "Chapter 6: Set Theory 6.1 Set Theory: Definitions and the Element Method of Proof 1 6.1 Set Theory - Definitions and the Element Method of Proof The introduction."

Similar presentations


Ads by Google