INDEX
Complex Numbers
Exponentiation
Algebraic Transformations
Algebraic Equations and Inequalities
Functions
Discrete Algebra
Basic Formulas
Graphics of Basic Functions
Numbers and Sets
Balloon

Introduction

About Algebra

Arithmetic Operations

Addition and Subtraction

Multiplication and Division

Criterions for Divisibility

Real Number System

Types of Numbers

Geometric Interpretation of Real Numbers

Irrational Number

Properties of Real Numbers

Fractions
Proportions

Property of Equal Proportions

Absolute Values

Graphical Illustrations

Sets and Intervals

Sets
Subsets
Operations with Sets
Intervals

Subsets
Key Topics Remaining: Intervals

A set  A  is equal to  B  if every element of  A  is an element of  B , and every element of  B  is an element of  A .

Examples of equal sets:

  • The sets  A = { a, b, c }  and  B = { c, a, b }  consist of the same elements, and so  A = B . 


  • Likewise,  { a, b, c } = { a, a, b, c}. 

    • The set  A  is said to be a proper subset of  B ,

    ,
    if every element of  A  belong to  B  but the set  B  contains some extra elements.

    Examples of proper subsets:

  • A null set is a subset of any set.


  • The set of natural numbers is a proper subset of the set of real numbers.


  • The set  A = { a, b, c }  is a proper subset of  B = { d, a, b, c, e },  since  B  contains all elements of  A  but there exist at least one element, which belongs to  B  but not  A .

    • A set  A  is a subset of  B , if  A  is a proper subset of  B  or  A  is equal to  B , that is,

    Examples of Subsets:


    • Examples
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