Sets are indeed an important method of representation of a group of data that satisfies a particular condition. Sets are categorised into two categories, i.e., finite and infinite sets. The word finite stands for something that can be counted, and infinite means something that is uncountable.
The properties and graphical representations of both finite and Infinite sets properties are completely different. One needs to understand this concept in-depth to be able to solve complex mathematical questions of sets.
Finite Sets
Sets that have a finite number of values in them are called finite sets. These sets are often referred to as countable sets because we can easily count the values in them. The number of values in a finite set is limited. This set contains all the values from the starting till the end that satisfies a particular condition. These sets can easily be written in a roster or tabular form.
For instance, a set consisting of the first six English alphabets is represented as
Set N = { a, b, c, d, e, f}
The aforementioned set is finite, as its values can easily be counted.
Infinite Sets
The set which consists of infinite numbers of values is called an infinite set. These sets are also called uncountable sets. It cannot be represented in roster form because it has an uncountable number of values present in it. Three dots represent the first and last limits of an infinite set.
For example:
- The set of natural numbers
- The set of all the real numbers
Finite Set’s Properties
There are several properties of finite sets.
- A proper subset of a given finite set will always have a finite value.
- A set formed due to the union of two finite sets is always a finite set.
- Set formed by the merging of two finite sets is always finite.
- The power set of a finite set is always finite.
- The cartesian product of any finite set is finite.
- Its cardinality will also be finite and is equivalent to the number of values in the finite set.
Infinite Set’s Properties
Let’s have a look at the various properties of infinite sets.
- Union of any two or more infinite sets is always infinite.
- The power set of an infinite set is always infinite.
- The super subset of the infinite set is infinite.
- The subset of a given infinite set may or may not be treated as infinite.
- Infinite sets can either be uncountable or countable, depending on the condition of the set.
Cardinality Of Set
The cardinality of a set refers to the number of values inside the particular set. If there is a set given, i.e., A, whose number of values is ‘X, then the cardinality of that set will be represented as n(A) = x.
Some more examples of the cardinality of sets are as follows:
The cardinality of set P of all the English vowels is 5 n(P) = 5.
The cardinality of set Q of the months in a year is 12 n(Q) = 12.
The cardinality of set R of all the even numbers from 1 to 10 n(R) = 5.
The values inside these sets are written in curly brackets or roster form.
Cardinality Of Infinite Set
The cardinality of an infinite set is represented as n(A) = ?. It is because the number of values in an infinite set is unlimited, and thus, they cannot be counted. Therefore, the cardinality of such sets is null or infinite.
Non-Empty Finite Set
Non-empty finite sets consist of a large number of values that cannot be written concisely. The number of values in a non-empty finite set is denoted with the number of values as n(A), which consists of natural numbers that satisfy the condition of the non-empty set.
For example – P = {a set of the number of children born in India}
Though it is very difficult to find the number of children born in India, at any point, it will indeed be a natural number. Hence, we say it is a non-empty finite set.
Let R be a natural number that is less than x. The cardinality of set R will be m.
This is because, R = { 1, 2,3,4,….. m }
X = { a1, a2, a3….an}
Y = { a : a1 ? R, 1 ? i ? m }
In this expression, i is an integer that lies between 1 and m.
Solved Questions
Question 1. Check whether the given Set of days in a week is finite or infinite. Also, state the cardinal value.
Answer. The given set is finite.
As the days of the week can be counted{ Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday }, it is a finite set.
In this set, there are seven values. So, the cardinal value of this set is 7.
Question 2. If Set A = { a, b,c,d,f} and Set B = {a, b,c, d, g, f}.
Find AUB and A?B. Also calculate the cardinal value of each of them.
Answer. Here set A and set B both are finite sets as they contain a limited number of values. So, the cardinal value of set A is 5, and of set B is 6. Now, AUB = {a, b,c, d, e, f, g} and, A?B = {a, b, c,d,f } , n(AUB) = 6, a(A?B) = 5.
Conclusion
Finite sets contain values that can easily be counted and whose cardinality is never infinite or null. Cardinality refers to the total number of values that are present in a particular type of set. Infinite sets, on the other hand, are sets that contain a range of values that cannot be counted. This is a basic difference between these two types of sets.
Finite sets can be represented in roster form, while an infinite set cannot. Although every finite set can be counted, non-empty finite sets cannot be counted. These sets contain values that we know exist and can be counted, but it is not possible to represent such a large array of values, so we use a range to denote such values.
Salman is a prolific environmental writer, and has authored more than 500 articles in reputed journals, magazines and websites. He is proactively engaged in creating mass awareness on renewable energy, waste management, sustainability and conservation all over the world.
Salman can be reached on salman@cleantechloops.com
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