Learn About Equal and Singleton Sets With Properties and Examples

Sets are defined in mathematics as a collection of objects whose elements are fixed and cannot be changed. In other terms, a set is a collection of data that does not pass from person to person. The elements in the set cannot be repeated, but they can be written in any sequence. Capital letters are used to represent the set.

 

Sets are primarily utilized in our daily lives to represent mass data and Alldata collection. Sets are categorized based on the number of elements they include. Unit sets, finite and infinite sets, null sets, equal and unequal sets, singleton sets, and other types of sets exist.

 

Let’s learn in detail about two very important types of sets among the ones aforementioned. Therefore, let’s understand Equal and Singleton Sets along with their properties and examples.

Equal Sets

Equal sets have the same cardinality and all of their members are equal. They are typically represented by a capital letter followed by the braces “. In other words, if two or more sets include the same items and have the same number of elements, they are deemed equal. As an example, consider Set A = {2,4,6,8} and B = {2,4,6,8}. Sets A and B are then regarded as equal because their elements and cardinality are the same.

Properties of Equal Sets

We now understand what equal sets are. Following that, we’ll look at some of its important properties to help with understanding and identification:

 

• The sequence of the elements has no bearing on the equality of the two sets.
• The cardinality or the total number of elements in equal sets is equal.
• When two sets are subsets of one another, the two sets are equal and the set notation is A ⊆ B and B ⊆ A. Therefore, A = B.
• Equal sets of elements must contain equal elements.
• The cardinal number is the same in the power set of equal sets.
• The property of having an equal number of elements is shared by equal and equivalent sets.
• All equal sets are equivalent sets, while the opposite is not true.

Singleton Set

A set with just one element is referred to as a singleton set. The singleton set has the form A={a} and is also known as a unit set. The null set and the set itself are subsets of the singleton set.

 

The singleton set has the formula A = {a}, where A represents the set and the little alphabet ‘a denotes a singleton set element. A singleton set is sometimes known as a unit set because it has only one element. A singleton set has one cardinal number. Singleton set has only two subsets: the empty set and the set itself with a single element.

 

The number of subsets of a given set that are singleton sets is equal to the number of items in the given set. There are 5 items in the set A = {a, e, i, o, u}. As a result, the set has five singleton sets, {a}, {e}, {i}, {o}, {u} which are subsets of the provided set.

Properties Of Singleton Set

Some of the most important properties of a singleton set are as follows.

 

• The singleton set contains only one element.
• A singleton set has one cardinality.
• The singleton set has two subsets.
• Every singleton set has a subset called the null set.
• A singleton set has two subsets: the null set and the singleton set itself.
• A singleton set’s powerset has a cardinal number of two.