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Chapter 1

Sets and Numbers

 
1.1 About Sets

 Many people collect various objects and they classify them, e.g. "coins of the 10th century", "French paintings", etc.
 
Since ancient history people classified the numbers in categories, e.g. "the prime numbers", "the even numbers", etc.
 
Categories, like the above or even groups of objects of the same kind or not, which can be distinguished, are called sets.

According to the mathematician Cantor:
 
"Sets are every collection of objects which derive from our experience or are our intellectual creations, that can be distinguished the one from the other and are well defined."
 
The objects that consist the set are called members of the set. In the above definition of sets the phrase "are well defined" means that the members of the sets can be recognized with certainty. For example we cannot talk about the set of big numbers since there is not a rule that can define whether a number is big or not.

To represent a set we use one of the letters of the Latin alphabet. If we want to denote that x is a member of the set A, we write  and we read " x is a member of A". On the other hand, if we want to denote that the x is not a member of A we write and we read  " x is not a member of A".

A set can be represented with the ways described bellow:

        For example the set of the keys of a calculator that can be used to type the number 0.033 is:
        A= { 0 , . , 3 }           For example the set of even numbers can be denoted as:
          A={ / x even number } with description
          or
         A={... , -4 , -2 , 0 , 2 , 4 , ... } with quotation

We say that two sets A and B are equal when they have exactly the same elements and we write A=B
For example if A={ 0 , 1 , 2 } and B={ 2 , 1 , 0} then A=B.

We say that a set A is a subset of B when every member of A is a member of B and we write .
For example if A={ 0 , 1} and B={1 , 0 , 2 } then .

Immediate consequences of these definitions are:
 
  • for every set A
  • If  and  then 
  • If  and  then A=B