SET THEORY

A set is a collection of well defined entities, objects or elements.

A set is described by listing elements, separated by commas, within brackets.

DESCRIBING A SET

For example:

A set of vowels of English Alphabet may be described as { a, e, i, o, u}

A set of even natural numbers can be described as { 2, 4, 6, 8, 10}



Note: 

The order in which the elements are writtem makes no difference. 

Also, repetition of an element has no effect. 

For example {1, 2, 3, 2} is the same set as {1, 2, 3}.



FINITE AND INFINITE SETS

Finite set: A set is called a finite set, if its elements can be counted and the process of counting terminates at a certain natural number say, 'n'

Example: {1,2,3,4,5}  ,  {1,2,3,4,.................up to 100}

Infinite Set: A set which is not finite or in other words, a set in which the process of counting does not terminate is an infinite set. 

Example: Set of natural numbers,

or {2,4,6,8,10........................} set of even numbers

or {1,3,5,7,9..........................} set of odd numbers


EQUAL SETS

Two sets A and B are said to be equal, if every element of A is a member of B, and every element of B is a member of A. If sets A and B are equal, we write A = B.


Similarly,

Unequal sets: When they are not equal or there exists at least one distinct element between these two sets.

Let A = {1,2,5,6} and B = {5, 6, 2, 1}. Then A = B because each element of A is an element of B and vice - versa


UNIVERSAL SET

There happens to be a set 'U' that contains all the elements under consideration. Such a set is called the universal set.

For example :

A = {1, 2, 3, 4, 5}, B = {4, 5, 6, 7, 8, 9}, We can say that they are both contained in their universal set, which is a set of natural numbers.

In plane geometry, the set of all points in the plane is the universal set.

MORE EXAMPLE

Given that U = {4, 5, 6, 7, 8, 9, 10, 11, 12} universal set, list the elements of the following sets.

a) A = {x : x is a factor of 72}
b) B = {x : x is a prime number}
c) C = {x : x is a odd number}

solution: 

The elements of sets A and B can only be selected from the given universal set U
a) A = {4, 6, 8, 9, 12}
b) B = {5, 7, 11}
c) C = {5, 7, 9, 11}