Complement of Set: Definition, Properties with examples & applications

The Complement of set is a set that contains all the elements that are not in the original set. In other words, it is the set of all elements that belong to the universal set but do not belong to the original set. The complement of a set is denoted by a prime symbol (')

Complement of Set examples:

let's consider the set of even numbers: {2, 4, 6, 8, 10,12..} The complement of this set, denoted by (even numbers)', is the set of all elements that are not in the set of even numbers. In this case, the universal set is the set of all integers, denoted by Z. Therefore, (even numbers)' = {1, 3, 5, 7, 9, 11,}, which is the set of all odd numbers.

Another example is the set of employees who work for a company. The complement of this set, denoted by (employees)', is the set of all individuals who do not work for the company. In this case, the universal set is the set of all people, denoted by P. Therefore, (employees)' = {students, retirees, unemployed individuals, etc.}.

It is important to note that the complement of a set depends on the universal set under consideration. For example, if the universal set is the set of real numbers, then the complement of the set of natural numbers would include all the numbers except natural numbers. 

The complement of a set can also be expressed using set difference. That is, if A is a subset of the universal set U, then A' = U - A. For example, if A = {1, 2, 3} and U = {1, 2, 3, 4, 5}, then A' = {4, 5}.

In summary, the complement of a set is a useful concept in mathematics and has practical applications in various fields, including data analysis, probability theory, database systems, network security, and set theory.Let’s look at complement of set in detail…..

The complement of set A, denoted by A’, is the set of all elements in the universal set that are not in A. It is denoted by A’

Some Properties of Complement Sets

1) A ∪ A′ = U

2) A ∩ A′ = Φ
3) Law of double complement: (A′) ′ = A
4) Laws of empty set and universal set Φ′ = U and U′ = Φ.

Example1: If A = {1, 2, 3, 4} and U = {1, 2, 3, 4, 5, 6, 7, 8} then find A complement (A’).

Solution: A = {1, 2, 3, 4} and Universal set = U = {1, 2, 3, 4, 5, 6, 7, 8}

Complement of set A contains the elements present in universal set but not in set A. Elements are 5, 6, 7, 8.

∴ A complement = A’ = { 5, 6, 7, 8}

Example 2: If B = { x | x is a book on Algebra in your library} Find B’.

Solution: B’ = { x | x is a book in your library and x ∉ B }

Example 3: If A = { 1, 2, 3, 4, 5 } and U = N , then find A’.

Solution: A = { 1, 2, 3, 4, 5 }

U = N
⇒ U = { 1, 2, 3, 4, 5, 6, 7, 8, 9,10,… }
A’ = { 6, 7, 8, 9, 10, … }

Example 4: If A = { x | x is a multiple of 3, x ∉ N}. Find A’.

Solution: As a convention, x ∉ N in the bracket indicates N is the universal set.
N = U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,11, … }
A = { x | x is a multiple of 3, x ∉ N }
A = { 3, 6, 9, 12, 15, … }
So, A’ = { 1, 2, 4, 5, 7, 8, 10,11, … }

Another example of a complement of set can be seen in the context of a Venn diagram. Let's say we have two sets A = {1, 2, 3, 4} and B = {3, 4, 5, 6}.

We can represent these sets using a Venn diagram as follows:

Venn diagram

In this diagram, the area inside the circle representing A represents the elements of A, and the area inside the circle representing B represents the elements of B. The overlap of the two circles represents the elements that are in both A and B.

The complement of A is the set of all elements that are not in A, which is everything outside of the circle representing A. This is represented by the shaded area (blue color region) outside of the circle:

So, the complement of A is:

A' = {5, 6}

This represents the elements that are not in A. These are a few variations of the complement of a set, which can be useful in different contexts. Here are some common variations:

Absolute complement:

The absolute complement of a set A, denoted by A', is the set of all elements that do not belong to A, relative to the universal set U. That is, A' = {x ∈ U : x ∉ A}. For example, if A = {1, 2, 3} and U = {1, 2, 3, 4, 5}, then A' = {4, 5}

Relative complement:

The relative complement of set A with respect to a set B, denoted by B\A, is the set of all elements in B that do not belong to A. That is, B\A = {x ∈ B : x ∉ A}. For example, if A = {1, 2, 3} and B = {1, 2, 3, 4, 5}, then B\A = {4, 5}.

Complement in a topology:

In topology, the complement of a set A in a topological space X, denoted by X\A, is the set of all elements in X that do not belong to A. This is different from the absolute complement, which is defined relative to a universal set. For example, if X is the set of real numbers and A = (0, 1),

then X\A = (-∞, 0] U [1, ∞).

Complement in a Boolean algebra:

In Boolean algebra, the complement of a set A, denoted by Ā, is the unique complement of A with respect to the Boolean operations of union and intersection. That is, Ā is the unique set that satisfies A ∪ Ā = U and A ∩ Ā = ∅, where U is the universal set. For example, if A = {1, 2, 3} and U = {1, 2, 3, 4, 5}, then Ā = {4, 5}.

Complement of set is also used in de morgan's law

Overall, the different variations of the complement of set can be useful in different contexts, depending on the type of set and the mathematical or theoretical framework being used.

The complement of a set has several practical applications in various fields

Here are a few examples:

Data analysis: In data analysis, the complement of set can be used to identify outliers or anomalies in data structure. For instance, the complement of a set of normal data can be used to identify data points that fall outside the normal range, which could indicate an error in the data str an unusual occurrence.

Information retrieval: In information retrieval, the complement of a set can be used to retrieve documents that do not contain certain keywords or phrases. For example, if a user searches for "cats" but wants to exclude documents that contain the word "dogs", the complement of the set of documents containing the word "dogs" can be used to retrieve only "cats" related documents.

Computer programming: In computer programming, the complement of a set can be used to perform set operations such as union, intersection, and difference. For example, the complement of the set of users who have already registered for a service can be used to identify potential new users who have not yet registered.

Genetics: In genetics, the complement of a set can be used to identify genes or genetic traits that are not associated with a particular condition or disease. For example, the complement of the set of genes associated with breast cancer can be used to identify genes that are not associated with the disease, which could help identify potential new targets for treatment.

Overall, the complement of a set is a useful concept with a variety of practical applications in many fields.

Here are some important things to learn and understand about the complement of a set, both in theory and practice

In theory:

Definition: The complement of a set is defined as the set of all elements that do not belong to the original set. It is denoted by A' or Ā, depending on the context.

Properties: The complement of a set has several important properties, such as De Morgan's law, which state that the complement of a union of sets is the intersection of their complements and vice versa. Other properties include the fact that the complement of set is the original set, and that the complement of the empty set is the universal set.

Types of complements: There are different types of complements, depending on the context. The absolute complement is defined relative to a universal set, while the relative complement is defined relative to another set. The complement in a topology is defined in the context of a topological space, while the complement in a Boolean algebra is defined with respect to the Boolean operations of union and intersection.

In practice:

Calculation: To calculate the complement of a set, you need to identify the elements that do not belong to the set, either by listing them explicitly or by using a rule or condition. This can be done using various techniques, depending on the specific context and problem.

Applications: The complement of a set has many practical applications in various fields, such as probability theory, data analysis, database systems, and network security. Understanding how to use the complement of a set in these contexts can help you to solve problems and make decisions.

Visual representation: The complement of a set can be represented visually using Venn diagram or other types of diagrams. This can help to illustrate the relationships between sets and their complements, and to identify patterns or trends.

Overall, learning and understanding the complement of set involves both theoretical and practical aspects. By mastering the concepts and techniques related to the complement of a set, you can develop strong mathematical and problem-solving skills that can be applied in various contexts.