Finite Set and Infinite Set

Finite sets and infinite sets are diametrically opposed. The finite set, as the name implies, is countable and has a finite number of members. The set that is not finite is referred to as the infinite set. The number of items in an infinite set is not finite and can go on indefinitely. Please keep in mind that countable infinite sets, such as the set of rational numbers, can exist. In our daily lives, we encounter a variety of limited and infinite sets.

In this post, we will look at finite and infinite sets, as well as their definitions and attributes. We will also learn the distinction between finite and infinite sets with the assistance of examples.

Finite set and infinite set

Sets with a finite or countable number of items are called finite sets. It is sometimes referred to as countable sets since the components included inside them may be tallied. The process of counting items comes to a stop in the finite set. The set contains beginning and finishing components. In roster notation, finite sets are easily expressed. The set of vowels in English alphabets, Set A = a, e, i, o, u, is a finite set since the set’s constituents are finite.

Sets that are not finite are referred to as infinite sets. The constituents of infinite sets are, in fact, infinite. We can call a set infinite if it is limitless from beginning to end or has continuity on both sides. The set of whole numbers, W = 0, 1, 2, 3,…….., for example, is an infinite set since the number of elements is infinite. Uncountable infinite sets include the set of real numbers. The elements of an infinite set are represented by dots, which indicate the set’s infinity.

Properties of finite set and infinite set

Finite Set Properties

Now that we understand the notion of finite sets, let us look at some of their properties:

• A finite set’s appropriate subset is finite

• Any number of finite sets can be joined to form a finite set

• It is finite to intersect two finite sets

• Finite sets have a finite cartesian product

• A finite set’s cardinality is a finite number equal to the number of items in the set

• A finite set’s power set is finite

Infinite Set Properties

Let us go through some of the key features of infinite sets:

• An infinite set is the union of any number of infinite sets.

• An infinite set’s power set is unlimited.

• An infinite set’s superset is also infinite.

• An infinite set’s subset may or may not be infinite.

• Countable or uncountable infinite sets exist. The set of real numbers, for example, is uncountable, but the set of integers is countable.

Differences of finite and infinite set

Finite sets have a defined number of components, can be counted, and can be expressed in roster form. An infinite set is a non-finite set; infinite sets may or may not be countable. This is the fundamental distinction between finite and infinite sets.

An infinite set is one with no elements that can be enumerated. An infinite set has no last element. An infinite set is a set that may be matched one-to-one with a proper subset of itself.

Conclusion

Sometimes we may just say “countable” to signify “countably infinite.” However, to emphasise that we are omitting finite sets, we typically use the word countably infinite. Countably infinite contrasts with uncountable, which represents a collection that is so huge that it cannot be numbered even if we continued counting indefinitely.