Sets are defined in mathematics as a collection of objects whose elements are fixed and cannot be changed. In other words, a set is a collection of data that does not pass from one person to the next. 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 include the Empty Set, Finite Set, Equivalent Set, Subset, Universal Set, Superset, Disjoint Set and Infinite Set and more. During calculations, each type of set has its unique significance. Sets are primarily utilised in our daily lives to represent bulk data and data collection.

Let’s define Sets and different types of sets with proper set theory symbols and examples in detail.

Types of Sets

Empty Sets

A set with no elements is also known as a null set or void set and is denoted by symbol {}.

For example,

Set A = {a: a is the number of students studying in Class 5th and Class 6th}. As we all know, a student cannot learn in two classes at the same time, hence set A is empty.

Singleton Sets

Singleton sets are collections of just one element.

For example,Set A = { 8 } is a singleton set.

Finite and Infinite Sets

A finite set has a finite number of members, but an infinite set has elements that cannot be calculated but have some figure or number that is huge to be precise in a set.

For example,

  • Set A = {3,4,5,6,7} is a finite set since it has a finite number of elements.
  • A set of all-natural numbers.

A = {1,2,3,4,5,6,7,8,9……} is a infinite set.

Equal Sets

Sets A and B are considered equal sets if every element of set A is also an element of set B and every element of set B is also an element of set A. It signifies that sets A and B have the same items and may be denoted as

A = B

For example, let A = {3,4,5,6} and B = {6,5,4,3}, then A = B

And if A = {set of even numbers} and B = { set of natural numbers}, then A ≠ B, because natural numbers include all positive integers beginning with 1, 2, 3, 4, 5, and so on, whereas even numbers begin with 2, 4, 6, 8, and so on.

Subsets

A set X is said to be a subset of set Y if its elements are members of set Y, or if each element of set X is present in set Y. A subset of a set is represented by the symbol (⊂) and is written as X ⊂ Y.

You can alternatively write the subset notation as follows:

X ⊂ Y if p ∊ X ⇒ p ∊ Y

Power Sets

Power sets are the sum of all subsets. We already know that the empty set is a subset of all sets and that each set is a subset of itself. As an example, consider the set X = {2,3}. According to the aforementioned statements,

{} is a subset of {2,3}

{2} is a subset of {2,3}

{3} is a subset of {2,3}

{2,3} is a subset of {2,3} as well

As a result, the power set of X = {2,3},

P(X) = {{},{2},{3},{2,3}}

Universal Sets

A universal set is one that contains all the components of all other sets. It is commonly represented as ‘U.’

For example,

Set A = {1,3,5}, Set B = {2,3,4,5}, and C = {5,6,7,8,9}.

The universal set is therefore written as U = {1,2,3,4,5,6,7,8,9,}.

Disjoint Sets

Two sets X and Y are said to as disjoint sets if they have zero(0) as the result of their intersection and do not share any elements. This can be written as X ∩ Y = 0.

Also Read : Mathematics : A tricky subject, Solution of pair of two lines

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