Finite and Infinite Sets

Among the different types of sets, finite and infinite sets are the two that are generally considered while dealing with mathematics. Both of them are self-explanatory with their names; while ‘finite’ explains it to be countable and ‘infinite’ means uncountable. As we get deeper into this discussion, you would be able to understand things much better.

Finite Set

Finite sets are the ones with countable or finite number of members. Finite sets are therefore, known as countable sets also. If the set elements consist of finite members, the process would soon run out of elements as it proceeds further.

Example of finite sets

  • Q={ 0, 3, 6, 9, …, 99}
  • N={ c : c is an integer, 1<c<10}
  • The set of English alphabets (since it is countable).

Cardinality of finite sets

Suppose ‘a’ is known to represent the number of elements in the Set A, the cardinality of this finite set would be n(A)=a. The cardinality of the set of English alphabets would therefore be 26 since the total number of elements is 26.

Thus, n (A) = 26.

This way the elements in finite sets can be listed in the roster form and curly braces.

Properties of Finite Sets

Listed below are several finite set conditions that are always finite

  • A subset of a finite set.
  • The power set of any finite set and;
  • The union for two finite sets.

Infinite Set

If a set isn’t finite, it would be considered and addressed as an infinite set since the number of elements for these sets would not be countable and hence these cannot be represented in roster form. Infinite sets are thus also known as uncountable sets. Therefore, while presenting an infinite set, three dots (ellipse) are used to mark its infinity.

Cardinality of Infinite Sets

The cardinality of a considerable set is n (A) = b. Here, b is the number of elements in the set A. the cardinality of an infinite set would be infinite since the number of elements within it is infinite.

Properties of Infinite Sets

  • The union for two infinite sets would be infinite.
  • The power set for an infinite set would be infinite.
  • The superset of an infinite set would also be infinite.

Since the number of elements is unlimited, the power set of an uncountable set is likely to be infinite. Thus, you now have probable elements to distinguish between the finite and infinite sets.


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