In this chapter we will learn the difference between finite and infinite sets with properties and examples.
Finite Set definition
A set is said to be finite set if its elements can be counted by natural numbers.
This mean that the number of element of finite set is finite and gets over after certain point.
Void set ( which does not contain any element ) is also classified under finite set.
Number of element in finite set
The number of distinct element present in given set A is called order of set A or cardinal number and is denoted by n (A).
For example;
If there are three distinct element in set A then Order of Set n(A) = 3
Examples of Finite set with cardinal number
(i) Y = { x : x is letter in English Alphabet }
We know that in English Alphabets there are 26 letters.
Hence, the set Y is a finite set as it contains finite number of elements.
The Order of set Y is n(Y) = 26
(ii) A = { x : \mathtt{x^{2} =\ 36} }
Solving the expression \mathtt{x^{2} =\ 36} ;
\mathtt{x^{2} =\ 36}\\\ \\ \mathtt{x\ =\ \sqrt{36}}\\\ \\ \mathtt{x\ =\ +6\ ,\ -6}
The set A can also be represented as;
A = { 6, -6 }
Hence, the set A is a finite set as it contain two elements.
Order of set n(A) = 2
(iii) Y = { x ; x 𝜖 N & x < 100 }
The above set states that it contains variable x such that;
⟹ x belongs to natural number
⟹ x is less than 100
Hence, all the numbers from 1 to 99 are part of set Y.
The set Y can also be expressed as;
Y = { 1, 2, 3, 4, 5 . . . . . 99 }
So the set Y is a finite set as it contains finite number of elements which ends after certain point.
Also, the order of set Y is 99.
n (Y) = 99
(iv) P = { persons living in New Zealand }
Since there are finite number of people living in New – Zealand and can be easily counted, the above set P is a finite set.
Infinite Set definition
A set which contains unending number of elements are called infinite set.
Since the number of infinite set are unending, it is not possible to count the element and give order of the set.
Examples of Infinite set
(a) A = { Number of stars in Universe }
There are infinite numbers of stars in universe. Hence the given set A is infinite set.
(b) X = { Set of even natural numbers }
The set X is an infinite set as it contains unending number of elements.
The set X can also be expressed as;
X = { 2, 4, 6, 8, 10, 12, 14, 16, 18 . . . . }
You can see that the number of elements goes on & on without any end. Hence it’s a perfect example of infinite set.
(c) Z = { set of all points in a plane }
In a plane, there can be infinite number of points.
Hence, the set Z is an infinite set.
(d) Y = { x ; x 𝜖 N & x > 5 }
The set states that it contain variable x such that;
⟹ x belongs to natural number N
⟹ x is greater than 5
The above set Y can also be expressed as;
Y = { 6, 7, 8, 9, 10, 11, 12 . . . . }
You can see that set Y contains unending number of elements. Hence the given set is infinite set.