A set is a well-defined collection of elements, that is, it must be clear whether a given element
belongs to the set or not.

Usually, sets are denoted by the capital letters A, B, C, etc., and elements of the sets by lower case letters a, b, c etc. If A is a set and a is an element of A we use the notation a ? A, if b is not an element of A then we write b ? A.

The empty set is unique.The empty set Ø is a subset of any set. Note that the empty set is a subset of any set A and A is its own subset: Ø ? A, A ? A. . If the set A is not a subset of the set D, this relation is written as A ? D..

The set of all elements of the sets A and B is called the union of the sets A and B and is denoted by A ? B.Union is the act of combining two sets together into a single set.

The set of elements A which are not in B is called the difference between A and B and is denoted by A \ B. A = {3, 5, 8, 10}, B = {4, 5, 9}. Then A \ B = {3, 8, 10}.

Usually the sets that we deal with are subsets of some ambient set. Such a set is called a universal set and is denoted by U. In other words, U is the universal set if all the sets under examination are subsets of U. The difference U \ A is called the complement of A and is denoted by Ac . That is,
Ac = U \ A = {x| x ? U and x ? A}

The intersection of two sets A and B, written A ? B is the set consisting of the elements of both A
and B. Thus, A ? B = {x| x ? A and x ? B}

A = {1, 2, 3, 5, 8}, B = {2, 3, 10, 11}. Then A ? B = {2, 3}.

Two sets A and B are called disjoint if A ? B = Ø, that is their intersection is the empty set.

A set A is called finite if it consists of a finite number of elements. Otherwise it is called an
infinite set.The set A = {1 ,3, 5} is finite, whereas the set B = {1, 3, 5, 7, …} is infinite.
For a finite set A, the number of elements in this set is denoted by s(A). For A = {1, 3, 5} we have s(A) = 3. The following equality is true:
s(A ? B) = s(A) + s(B) –s(A ? B)

A = {3, 5}, B = {1, 2, 3, 5}, C = {1, 2, 4, 6}. Then
A ? B, A ? B = A = {3, 5},
B ? C = {1, 2, 3, 4, 5, 6},
A ? C = Ø

Ø, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}.
The set A has 2x2x2 = 8 subsets.

The most familiar number set is the natural numbers, denoted by N= {1, 2, 3, 4, ...}Every natural number is an integer. Every integer is a rational number. We do not consider the zero as a natural number. The sum of two natural numbers is again a natural number, whereas the difference might not be a natural number.

A rational number is defined as a quotient of two integers with nonzero denominator and is
denoted by Q. A rational number is a fraction m/n where m and n are integers and n ? 0. Every rational number has an infinite number of representations by fractions. Rational numbers with denominators 10, 100, 1000, ... have special representations, named decimal representations. Every rational number has a finite or repeated infinitive decimal representation by using decimal fractions. 

A real number which is not a rational is an irrational number.Every irrational number has rational numbers arbitrarily close to it. The union of the sets of rational and irrational numbers is called the set of real numbers and is denoted by R .

3b = 4a, b = 4/3 · a. Since a < 0 then b < a. c = b/2 = 4/3· 2 · a = 2/3 · a.

Therefore c > a, and b < a < c.

Powers are used when we multiply a real number by itself repeatedly. For a ? R and n ? N , we define a to the power n.For a ? 0, the zero exponent and the negative exponents are defined as follows:a⁰ =1 a^-n = 1/aⁿ . If n is an even natural number then nth root of a, an , is defined
only for nonnegative numbers a. If n is odd then an is defined for all numbers a.

(a ·b)ⁿ = an ·bn

(a/b)ⁿ = aⁿ/bⁿ   (b? 0)

a^m · aⁿ = a^m+n

(a^m)ⁿ=a^m+n

Intervals are important subsets of the real numbers. Given two real numbers a and b with a < b the set {x| x ? R and a ? x ? b}is called a closed interval and is denoted by [a, b]. Similarly, the half-open intervals (a, b], [a, b) are defined as (a, b] = {x| x ? R and a < x ? b}, [a, b) = {x| x ? R and a ? x < b}.The intervals, defined above are finite intervals. Using the symbols ? (plus infinity) and –? (minus infinity) unbounded intervals can be defined, namely

(a, ?) = {x| x ? R and x > a},

a, ?) = {x| x ? R and x ? a},

(–?, b) = {x| x ? R and x < b},
(–?, b] = {x| x ? R and x ? b}.

For a given real number a on the real line, the distance from a to the origin is called the absolute value of a and is denoted by |a|. The absolute value has the following properties:
|a| ? 0, |a + b| ? |a| + |b|, |a . b| = |a| .|b|, |a / b| = (|a|) / (|b|) (b?0) .
Open and closed finite intervals can be represented by using the absolute value.

The middle points are c = (-1+ 5)/2 = 2 and c = (4 + 6)/2 = 5 the lengths are b – a = 5 – (-1) = 6 and b – a = 6 – 4 = 2, respectively.

[-1, 5] = {x | x ? R , |x – 2| ? 3},
(4, 6) = {x| x ? R |x – 5| < 1}