Finite Sets and Infinite Sets

Sets with a countable or finite number of elements are called finite sets. Because the elements in them can be counted, they are also referred to as countable sets. The counting of elements ends in a finite set. Roster notation makes it simple to represent finite sets. For example, in the English alphabet’s vowel set, set A = {a, e, i, o, u}, is said to be finite since it has a limited number of elements.
 

If a set is uncountable, has no end or limit, then it is an infinite set. For example, the set of whole numbers, that is W = {0, 1, 3,.....}, is infinite because the number of elements in the set is uncountable; the set of real numbers is an example of infinite sets; and the elements of an infinite set are represented by dots because the dots indicate indefinite continuation of the sets whether countable or uncountable.

Difference between finite and infinite sets

Finite sets can be differentiated from infinite sets in the following ways:

Let’s talk about some characteristics of finite sets now that we understand the concept:

• A finite set has a proper subset that is also finite.
• The union of any number of finite sets is also finite.
• Two finite sets have a finite intersection.

What are the Properties of Infinite Sets

Let’s review some of the key characteristics of infinite sets:

• Any number of infinite sets can be joined together to form an infinite set.
• The power set of an infinite set is also infinite.
• An infinite set’s superset is said to be infinite.

How to Represent Finite and Infinite Sets in Venn Diagram 

A Venn diagram uses overlapping circles to show relationships with each other. Each circle inside is a set, and it is different from one another. This shows the relationship between different sets.

For example, in the Venn diagram above. It includes:
P = {a, b, c, e, i}
Q = {e, i, d, f, h}.
P ∪ Q = {a, b, c, e, i, d, f, h}
P ∩ Q = {e, i}
The results of the union of sets (P ∪ Q) and the intersection of sets (P ∩ Q) are finite, because the sets P and Q are finite, that is, n(P) = 5 and n(Q) = 5.
Now look at the Venn diagram below:

Two sets are shown in the diagram: an infinite set represents a set of whole numbers (which is the violet part), and the inner set (the pink part), which contains the data {3, 8, 13}, represents a finite set. Here, the set contains an infinite number of elements, so this is said to be infinite.

How to know if a Set is Finite or Infinite?

We are aware that a set is referred to as finite when the elements in the set are countable. Let us examine a few criteria to determine whether a set is infinite or finite:
Unlike a finite set, an infinite set can have continuity from both ends and has no end points. We have both the beginning and the ending elements in a finite set. An infinite set is one whose elements cannot be counted, while a finite set is one whose elements can be counted.
 

Finite sets have countable elements, and infinite sets are not countable. Let us know how finite and infinite sets are used in daily life.

• Management of inventory
  A finite set, such as S = {rice, sugar, oil, soap, shampoo}, is used by a shopkeeper to keep track of items. It facilitates inventory management, reordering, and sales report management because the list of available products is small and countable.
• Attendance of students
  A teacher uses a set such as A = {John, Meera, Ravi, Priya} to keep track of the students who are present. Due to the fixed class size, this set is finite, which facilitates daily attendance tracking.
• For tracking websites
  V = {v1, v2, v3, …}, where each vn represents a user. Since there is no upper limit to the number of users that can visit, this creates an infinite set that is helpful for trend analysis and resource scaling. And since the potential number of users is not fixed, the set becomes infinite.
• An infinite set of real numbers
  T = {t | t ∈ ℝ, t ≥ 0} is used to measure time. This infinite set can be used to create continuous timelines for events, physics computations, or project planning.
• Library book categories
  To effectively classify, store, and retrieve books, a librarian groups them into categories such as C = {fiction, non-fiction, biography, science, history}.