Chapter 1

1.1 SETS DEFINITION OF SET METHODS FOR SPECIFYING SET SUBSETS VENN DIAGRAM SET IDENTITIES SET OPERATIONS

1.1.1DEFINITION 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

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

1.1.2METHODS 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: 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.

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

1.1.3SUBSETS:  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.

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

1.1.4VENN DIAGRAMS:  P  Pictorial representation of set in which sets are represented by enclosed areas in the plane.  The universal set U is represented by the interior of a rectangle.  The other sets are represented by disks lying within the rectangle. U B A

1.1.5SETS OPERATIONS : Symmetric Symmetric Union Difference Intersection Complement Disjoint Difference Disjoint Difference 1) 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

Find the union of the sets.

2) 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

Find the intersection of the sets;

Properties of Union and Intersection  Property 1:  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:  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,

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

Suppose sets. Find.

4) 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

Find the difference of.

5) 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

Find: a) b)

6) 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

Find: a) b) c)

1.1.6SET IDENTITIES: Sets under operations of union, intersection, and complement satisfy various laws / identities which are listed in Table 1.