Introduction to Set Theory





Introduction to Set Theory:

The concept of modern mathematics is started with set. Set appears in all branches of mathematics. The main developer of set theory is George Cantor (1845-1915 AD).

George Cantor

George Cantor

The word set is synonym with “Collection” , “Class” or “Aggregate”. Basically set is a collection or organization of similar objects (an object may be material or conceptual). and the objects by which a set is made of  are called elements or member of the set. Some example of set are:
1> The countries of Europe.

2>The solar system.

3>The peoples living in my house.

4>Vowels of the English alphabets. etc.

Notation:

Sets are usually denoted by capital letters and the elements of set are denoted by small letters. For example : Set A={a,b,c,d} The symbol  \in  denotes set membership and symbol \notin   denotes non membership. For example in set A above, a  \in A but e \notin A , Which means the element “a” belongs to set “A” but the “e” doesn’t.

Specification:

A set can be denoted or membership of a set may be indicated in several ways. Two common ways among them are:

a> Listing or Tebulation. In this method elements of set are listed , sperated by commas and enclosed inside a bracket. For example: V={a,e,i,o,u} My family={me , my son , my spouse}

b>Description or Rule. In this method a set is specified by enclosing in brackets a descriptive phrase or a rule. For example: V={the vowels of the English alphabet.} N={x:x is a natural number} A={x:x^2-3x=0} In last two examples the respective set is a group of element x , where each value of element x is defined by the respective rule.

Finite and Infinite set:

If a set have a finite numbers of elements then a set is called finite set else the set is called infinite set. For example: D={0,2,4,6,8} A={a,e,i,o,u} The above two sets are finite set while the following sets are infinite one. The set of stars in Universe. D={set of natural numbers}

Null Set:

A set which has no element is called Null set. A Null set is also called Empty set or Void set. It is denoted by symbol ø  For example: A={x:x is a man who gave birth to a child) N={x:xis a natural number , x>0 and x<1 }

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