Sets-Types of Sets

The number of elements in a set determines what type of set it is. Sets are groups of components of the same type—for instance, a collection of prime numbers, natural numbers, and so on. Unit sets, finite and infinite sets, empty sets, equal and unequal sets, and so on are all examples of sets. Let’s take a closer look at the various types of sets.

Different Types of Sets

A set is a well–defined collection of items, as we all know, depending on the nature of the objects and their properties. There are several sorts of sets. Some of these are discussed further down.

Singleton Sets or Unit Sets

A singleton set is a set that only has one element. Because it only has one element, it’s also known as a unit set. Set A is equal to {k | k is an integer between 5 and 7, resulting in A being equal to {6}.

Finite Sets

As the name implies, a finite set has a finite or precise number of members. It’s called a non-empty finite set if the set isn’t empty. The following are some examples of finite sets: Set B = {k | k is an even integer less than 20}, for example, is B = {2,4,6,8,10,12,14,16,18}. Consider the following example: Set A = {x: x is a day in a week}; Set A will have 7 entries and hence,a finite set.

Infinite Sets

An infinite set is one of the popular types of sets containing an unlimited number of items. To put it another way, if a set is not finite, it will be endless. For example, A = {x: x is a real number};, and real numbers are infinite. As a result, A is an infinite set. Let’s look at another case. Set B = {z: z is the coordinate of a point on a straight line}; is the coordinate of a point on a straight line; a straight line has an infinite number of points.

Empty Set

An empty set, also known as a null set, is a set that has no elements. The symbol’∅’ is used to represent an empty set. It’s pronounced ‘phi.’ Set X = {} as an example.

Sets of Equals

When two sets contain the same items, they are equal sets. A = {1,3,2} and B = {1,2,3} are two examples. Sets A and B are equal in this case. A = B can be used to represent this.

Unequal Sets

Unequal sets are those that have at least one different element. X = {4, 5, 6} and Y = {2,3,4} are two examples. Sets X and Y are unequal sets in this case. X ≠ Y can be used to symbolise this.

Equivalent Sets

When two sets contain the same number of elements but distinct elements, they are equivalent sets.

Overlapping Sets

If at least one element from set A appears in set B, the two sets overlap. A = {4,5,6} B = {4,9,10} is an example. Element 4 appears in both sets A and B in this case. As a result, A and B are two sets that overlap.

Disjoint Sets

If there are no shared elements in both sets, they are disjoint sets. Example: A = {1,2,3,4} B = {7,8,9,10}. Here, disjoint sets are set A and set B.

Subset and Superset

If every member in set A is also present in set B, set A is a subset of set B(A ⊆ B) and set B is the superset of set A(B ⊇ A).

Example: A = {1,2,3} B = {1,2,3,4,5,6}

Because all the components in set A are also present in set B, A ⊆ B.

Set B is the superset of set A, denoted by B ⊇ A.

Universal Set

Another type of set is the universal set, a collection of all items related to a specific topic. The letter’ U’ signifies a universal set in set notation. Let U stand for “the list of all road transport vehicles.” This universal set includes a set of automobiles, a set of cycles, and a set of trains, all of which are subsets of this universal set.

Power Sets

The collection of all subsets that a set might contain is called a power set.

Conclusion

Set theory is an area of mathematics that studies the properties and types of sets. Set theory’s fundamental notions are straightforward and appear to be self-evident. Despite its apparent simplicity, set theory turns out to be a tremendously complex subject. Mathematicians have demonstrated that the theory of sets can codify almost all mathematical notions and outcomes.