Sets

A set is a well-defined collection of elements.

Sets are usually denoted by the letters A,B, C,..., and the elements of the sets are represented by the small letters a ,b, c . If the element belongs to the set A , we denote this fact by use of the notation31, if the element does not belong to then we write31.

A set can be defined by listing all elements or by giving a rule that determines all the elements of the set. For example,

31

The set Ais a subset of the set B , written31, if every element ofA is also an element ofB. Two sets and are equal, written31, if both inclusions 31and31 are satisfied simultaneously.

The set containing no element is called the empty set and is denoted by 31 . The set 31is a subset of any set.

Example: 31,31,31 Then 31and31

It is important to represent a set by writing its elements inside a closed planar curve. Such a representation is called a Venn diagram.

31

Venn diagrams of set A={2,3,6,8}.

31

Venn diagrams of setB={2,5,8,9,10}.

If in some problems, all the sets under examination are subset of a wider set then this wider set is called a universal set and usually is denoted by U . For example, if in some problem all sets are subsets of the set of natural numbers not greater than then the universal set becomes

31

-The set31 is called the union ofA andB,

-The set31is called the intersection ofA and B ,

-The set31 is called the difference of Aand B .

The set is called the complement of A , where Uis the universal set.

31

The Venn diagram for 31.

31

The Venn diagram for31.

31

The Venn diagram for31.

Example: 31,31,31. Then

31

A setA is called finite if it contains finite number of elements. Otherwise, it is called an infinite set. For a finite setA, the number of elements in this set is denoted by s(A) . For given finite sets Aand B , the equality

31

is true. If A hasn-elements then the total number of all subsets is31.

Properties of Set Operations

Set operations have many obvious properties. We list some of them below.

i.31

ii. 31

iii.31

iv.31

v.31

vi.31

Numbers

The most familiar number set is the set 31of the natural numbers:

31

The set of all integers is denoted by

31

A quotient of two integers with nonzero denominator is called a rational number. The set of all rational numbers is denoted by31.

Rational numbers with denominators 10 , 100 , 1000, ...are called decimal fractions. For example,

31

are very simple examples of decimals numbers.

Every rational number has finite or repeated infinite decimal representation. The real line is a line with a fixed scale, origin and direction.

Every rational number has a unique representation on the real line. On the other hand, there are infinitely many points on the real line which cannot be represented by rational numbers. The numbers, corresponding to such points are called irrational numbers. For example,31,31,31 are irrational numbers.

31

The union of the sets of rational and irrational numbers is called real numbers and is denoted by31.

Inequalities

Let31 and 31be real numbers and these numbers correspond the points and on the real line. If the pointA lies to the left of the pointB then it is written as31 or 31 a ( a is less thanb orb is greater thana). If ais not greater than then it is written as 31or31. These relations are called inequalities and below we give the most important properties of inequalities:

-If31 then31

-If31 and 31then31, if and31 then 31 .

In order to list fractions in some order (increasing or decreasing) the fractions should be reduced to a common denominator or numerator.

1) Given 31 , 31 ,31,31, the number 31is a common denominator.

31

therefore

31

2) Given31, 31 , 31 , 31, the number31 is a common numerator

31

Therefore

31

Powers and Roots

Given31 and31, ato the power ofn is defined as31.

If 31then31, 31 .

For a given31, a number31 satisfying the equality31 is called then-th root of the number and is denoted by31 or 31 . If 31the symbol 31is used instead of31. Ifn is odd and 31then by definition 31 .

31

The powers and roots have the following properties. Let pandq be fractions with positive denominators. Then

31

Examples:

31

Intervals of Real Numbers

Intervals are subsets of R and are defined as follows:

31

For example 31

31

The set of all real numbers R is the interval31.

The absolute value of a real numbera, written|a|, is defined by

31

and has the following properties:

31

For example 31

Example: Represent the intervals(4,6),[-2,2] by using the absolute value.

Solution. The middle point of31

31

The middle point of[-2,2] is 0 , the length is 4, therefore

31