Finite Set

There are numerous categories of sets. This article concerns itself with the specificities of Finite sets and, in turn, infinite sets. But it also makes the general idea of sets and their basic categories clear. Set theory is a relatively new chapter in mathematics and is still in the process of building. The preliminary idea of a finite set is hidden in its name itself. We call a set finite when its elements are defined and countable. In other words, you will run out of elements eventually while constructing a finite set. Let us go deeper into set theory and finite sets to understand the idea better.

The Definition of a Set and its Categories

A set is a collection of elements defined by a common property. This collection of elements may be finite or infinite. There are some common symbols that represent much-used forms of the set. For example- N represents a set of all the natural numbers, Z represents a set of all integers, Q represents a set of all rational numbers, R does the same for all real numbers, etc.

The representation of sets has three forms:

• Roster Form: For example: A set of odd numbers less than seven= {1, 3, 5}
• Statement Form: For example :A= {all odd numbers less than 7}
• Set Builder Form: e.g, A=[ x|x, is an even natural number less than 15] this means A= {2, 4, 6, 8, 10, 12, 14} the values of x are all the elements in A, i.e., x has multiple values.

Now let us look at different categories of sets.

• Empty or Null Set

An empty set does not contain any elements. In other words, it can also be called an absurd set because the conditions it has cannot be satisfied with any element. For example, a set of months with seven Sundays or a set of dogs with five legs. These sets have no specific or possible answers. Empty sets or null sets are mostly signified with ɸ symbol.

• Subset

A subset is a set that belongs to another set. For example, if N={ a set of all the natural numbers} and A={1, 2, 3, 4, 5} then naturally, the set A belongs in the set N. In simpler words, A is a subset of N. N, in turn, is called the superset of A. A set can have multiple subsets or can belong to multiple supersets.

• Universal Set

A universal set contains all of the elements present in other sets while constructing a Venn diagram.

• Power Set

A power set is a set that contains all the subsets of a specific set. It is usually signified by the letter P.

What are Finite Sets?

The key to understanding finite sets is there in its name. A set that contains a finite or countable number of elements can be called a finite set. Usually, while solving problems of set theory at the primary level, we deal with finite sets.

Example

An empty set can be considered a finite set since we can count the number of its elements. A more generic example of a finite set would be–

A={1, 2, 3, 4, 5}, or

A={ months that have 31 days}

A= {The number of holidays in a specific calendar}

What are Infinite Sets?

Infinite sets are sets that have uncountable or infinite elements.

Example

A= {1, 2, 3,…}: A set of all rational numbers, a set of all integers,  or a set of all the natural numbers.

Properties of Finite and Infinite sets

Properties of Finite Sets

The subset of a Finite set will always be a finite set.

A Finite set has to be a union of two other finite sets

A finite set has to be the power set of another finite set.

The cardinality of a finite set is defined or countable [ the cardinality of a set is the number of elements that a set can have, it is signified by n(A) when A is a set].

Properties of an Infinite Set

When two infinite sets are united, the union results in an infinite set.

An infinite set has to be a power set and a superset to only another infinite set.

The cardinality of an infinite set is immeasurable.

An infinite set can have continuity in both ends of the set, i.e., an infinite set does not have a start or an end.

How to Identify a Finite Set

• You can look for a continuity marker like ‘…’. If a set has continuity markers on one end or both ends, it is an infinite set. This is done because it is hard to represent an infinite set in roster form without the continuity markers.
• Finite sets are easy to identify since you can count the elements. But finite sets can be large, so counting the elements can be a little inconvenient at times. If a finite set is presented in roster form, it is easy to understand.
• Infinite sets are not represented in a Venn diagram because the elements cannot be bound in a circle.

Conclusion

Sets have numerous categories, two of which are finite and infinite sets. Finite sets are those which contain a countable number of elements and are therefore called countable sets as well. These sets are easier to deal with in a Venn diagram since its counterpart, the infinite set, cannot be represented through a Venn diagram. The above article is an extensive and comprehensive study on finite sets that deals with interesting properties of finite sets and how to identify one.