CHAPTER 3 SETS, BOOLEAN ALGEBRA & LOGIC CIRCUITS PART I –SETS

CHAPTER OUTLINE: PART i 1.1 SETS 1.1.1 DEFINITION OF SET 1.1.2 METHODS FOR SPECIFYING SET 1.1.3 SUBSETS 1.1.4 VENN DIAGRAM 1.1.5 SET OPERATIONS 1.1.6 SET IDENTITIES

3.1 Sets - empty set 1.1.1 DEFINITION OF SET: Unordered collection of distinct objects and may be viewed as any well-defined collection of objects called elements or members of the set . Notations: Usually uses capital letters, A, B, X, Y to denote sets. lowercase letters, a, b, x, y to denote elements of sets. - denote x is an element of set A. - denote x is not an element of set A. - empty set

Example 3.1 If G is the set of all even numbers, then : Special Symbols: N = the set of natural numbers or positive integers: 1, 2, 3,... Z = the set of all integers: ..., -2, -1, 0, 1, 2,... Q = the set of rational numbers R = the set of real numbers C = the set of complex numbers

3.1.2 METHODS FOR SPECIFYING SETS: 1) Listing: by listing its elements between curly brackets { } and separating them by commas. E.g: A = {0}, B = {2, 67, 9}, C = {x, y, z}. 2) Set’s Construction / Implicit Description: by giving a rule which determines if a given object is in the set or not. E.g: A = { x : x is a natural number} B = { x : x is an even integer, x > 0} C = { x : 2x = 4} We describe a set by listing its element only if the set contains a few elements; otherwise we describe a set by the property which characterizes its element.

Example 3.2 List all the elements of each set when N = {1, 2, 3, ...}. ii) iii)

3.1.3 SUBSETS: If every element of A is also an element of B. That is Written as Two sets are equal if they both have the same elements, or equivalently if each is contained in the other. That is: If A is not a subset of B, or at least one element of A does not belong to B, we write .

Subsets: Property 1: It is common practice in mathematics to put a vertical line “|” or slanted line”/” through a symbol to indicate the opposite or negative meaning of a symbol. Property 2: The statement does not mean the possibility that . In fact, for every set A we have since every element in A belongs to A. However, if and , then we say A is a proper subset of B (sometimes written ).  Property 3: Suppose every element of a set A belongs to a set B and every element of B belongs to a set C. Then clearly every element of A also belongs to C. In other words, if and , then .

Example 3.3 Let A = {2, 3, 4, 5}, a) Show that A is not a subset of b) Show that A is a proper subset of

A large square is used to represent the universe set U. 3.1.4 VENN DIAGRAMS: Pictorial representation of set in which sets are represented by enclosed areas in the plane. A large square is used to represent the universe set U. The element of the universe in a set S fall inside the circle for S, while elements not in the set S fall outside of that circle. U S

Intersection Complement Disjoint Difference Union 3.1.5 SETS OPERATIONS : Symmetric Union Difference Intersection Complement Disjoint Difference Union Let A and B be sets. The union of sets A and B contain those elements that are either in A or B, or in both ( ). Denoted: U A B

Example 3.4 Find the union of the sets .

Intersection Let A and B be sets. The intersection of sets A and B contain those elements in both A and B ( ). Denoted: U A B

Example 3.5 Find the intersection of the sets;

Properties of Union and Intersection Property 1: Every element x in belongs to both A and B; hence x belongs to A and x belongs to B. Thus is a subset of A and B; namely Property 2: An element x belongs to the union if x belongs to A or x belongs to B; hence every element in A belongs to , and every element in B belongs to . That is,

Disjoint Two sets are called disjoint if their intersection is the empty set ( ).

Example 3.6 Suppose sets . Find .

Also called the complement of with respect to . Difference Let A and B be sets. The difference of A and B is the set containing those elements in but not in . Denoted by Also called the complement of with respect to . U A B

Example 3.7 Find the difference of .

Similarly can be define as . Complement Let be the universal set. The complement of the set is the complement of with respect to . Denoted by Similarly can be define as . U A

Example 3.8 Find: a) b)

Symmetric Difference The symmetric difference of sets and consists of those elements which belong to or but not to both Denoted by . U A B

Example 3.9 Find: a) b) c)

3.1.6 SET IDENTITIES: Sets under operations of union, intersection, and complement satisfy various laws / identities which are listed in Table 1.

EXAMPLE

EXERCISE 1 2

EXERCISE 3 ) In a survey of 260 college students, the following data were obtained: 64 had taken a mathematics course, 94 had taken a computer science course, 58 had taken a business course, 28 had taken both mathematics and business course, 26 had taken both mathematics and computer science course, 22 had taken both a computer science and a business course, and 14 had taken all three types of courses. (a) How many students were surveyed who had taken none of the three types of courses? 106 (b) Of the students surveyed, how many had taken only a computer science course? 60

EXERCISE

The Inclusion-Exclusion Principle

EXERCISE