Chapter 1. 1.1 SETS 1.1.1 DEFINITION OF SET 1.1.2 METHODS FOR SPECIFYING SET 1.1.3 SUBSETS 1.1.4...

Chapter Chapter 11

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

1.1.6 SET IDENTITIES

1.1.5 SET OPERATIONS

1.1.11.1.1 DEFINITION OF SET: 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: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

x Ax A

or

If G is the set of all even numbers, then :

Special Symbols: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

4 G3 G

1.1.21.1.2 METHODS FOR SPECIFYING SETS: 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: 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)

3 9A x x

is even, 11B x x x

4 3C x x

1.1.31.1.3 SUBSETS: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 .

A B,a A a B

if and only if and A B A B B A

A B

Subsets: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 .

A B

A BA A

A B A BA B

A B B C A C

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

Ax

is evenB x x

1,2,3,...,8,9C

1.1.41.1.4 VENN DIAGRAMS:VENN DIAGRAMS: PPictorial 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.

Ax

U

BA

1.1.51.1.5 SETS OPERATIONS :SETS OPERATIONS :

SymmetricSymmetric

Union Union DifferenceDifference

IntersectionIntersection ComplementComplement

Disjoint Disjoint Difference Difference

1)1)UnionUnion 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:

Ax

A B

U

A

B

or A B x x A x B

Find the union of the sets .

Ax

1,3,5 and 1,2,3A B

2)2) IntersectionIntersection Let A and B be sets. The intersection of

sets A and B contain those elements in both A and B( ).

Denoted:

Ax

A B and A B x x A x B

U

A

B

Find the intersection of the sets;

Ax

1,3,5 and 1,2,3A B

Properties of Union and IntersectionProperties 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,

Ax

and A B A A B B

A B

A B

A BA B

and A A B B A B

3)3) DisjointDisjoint Two sets are called disjoint if their

intersection is the empty set ( ).

Ax

A B

Suppose sets . Find .

Ax

1,2 , 4,5,6 and 5,6,7,8A B C , and A B A C B C

4)4) DifferenceDifference 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 .

Ax

A B

A B

U

A

B

B A

and A B x x A x B

Find the difference of .

1,3,5 and 1,2,3

5)5) ComplementComplement Let be the universal set. The

complement of the set is the complement of with respect to .

Denoted by

Similarly can be define as .Ax

U

A

UA A U

'/ /cA A A

and A x x U x A

U A

Find:a)b)

Suppose = 1,2,3,... is the universal set. Let 1, 2,3,4 , 3,4,5,6,7

2,3,8,9 , 2,4,6,... is the set of even integers .

U A B

C E E

, , and A B E

, , and A B B C E A

6)6) Symmetric DifferenceSymmetric Difference The symmetric difference of sets and

consists of those elements which belong to or but not to both

Denoted by .

Ax

A BA B

and or and A B x x A x B x B x A

A B

U

A

B

Find:a) b)c)

Suppose = 1,2,3,... is the universal set. Let 1, 2,3,4 , 3,4,5,6,7

2,3,8,9 , 2,4,6,... is the set of even integers .

U A B

C E E

A B

A C

B C

1.1.61.1.6 SET IDENTITIES:SET IDENTITIES:

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

Ax

Ax