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Finite and Infinite Sets

Let A = {a, e, i, o, u} 
B = {x : x is a river on the earth} 
C = {x : x is a human being living on the earth}.

If we count the number of elements in A, we find the number of elements in A is 5.
It may be difficult for us to count the number of elements in B and C. But, we are sure that if we try to count the number of elements in B or C, i.e. the rivers and human beings living on earth are countable. A set which has distinct elements is said to be a Finite set. Note that finite sets contain a definite number of elements.

Now consider the set of natural numbers. How many elements does this set contain? Does the process of counting its distinct elements come to an end? No. We see that the number of elements in this set is not finite or countable. A set which is not finite is said to be infinite. Let us look at some more examples of infinite sets.


Illustration 1
Let L = {x : x is a line in a given plane}, then L is an infinite set.

Illustration 2
Let C = {x : x is a circle in a given plane}, then C is an infinite set.

When we represent a set in the roster form, we write all the elements of the set within brackets { } separated by commas. In case a set contains infinite number of elements, it is not possible to write all the elements within { }. To represent an infinite set in the roster form, we write a few elements of the set to clearly indicate the structure of the set, followed by three dots. For instance {1, 3, 5, 7, 9, ...} is the set of all odd natural numbers and {..., -3, -2, -1, 0, 1, 2, 3, ...} is the set of all integers.

Which of the following sets are infinite?
  1. The set of triangles in a plane.
  2. The set of natural numbers divisible by 1050.
  3. The set of points common to two non-parallel intersecting lines.
  4. The set of points on a circle.
  5. The set of natural number 500.
  6. The set of even numbers between 100 and 500.
  7. The set of prime numbers between 100 and 700.
  1. Infinite   
  2. Finite     
  3. Finite    
  4. Infinite    
  5. Finite    
  6. Finite   
  7. Finite

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