Complement of a Set: Definition, Symbol, Properties

Complement of a Set: A well-defined collection of objects or elements is known as a set. Any set consisting of all the objects or elements related to a particular context is defined as a universal set. It is represented by \(U.\) For any set A which is a subset of the universal set \(U,\) the complement of the set \(A\) consists of those elements, which are the members or elements of the universal set \(U\) but not of the set \(A.\)

Set Definition

Definition: A set is a collection of well-defined objects.

Here, the word ‘well defined’ means it should be possible to determine whether an object does or does not belong to a specific collection. 

The objects of the set are known as its elements or members. The symbol is used to mean “is an element of”, and the symbol is used to denote “is not an element of”.

Example: In set \(A = \{ a,b,c\} ;a,b\) and c are the elements of set \(A.\)

We write, \(a∈A,\) for \(“a\) is an element of \(A”.\)

\(b∈A,\) for \(“b\) is an element of \(A”.\)

\(c∈A,\) for \(“c\) is an element of \(A”.\)

And \(d \notin A,\) for \(“d\) is not an element of \(A”.\)

Operations on Sets

1. Union of Set

Let \(A\) and \(B\) be any two sets. The union of \(A\) and \(B\) is the set that consists of all the elements of \(A\) and all the elements of \(B,\) the common elements being taken only once. The symbol \( \cup \) is used to denote the union. Symbolically, we write \(AB\) and usually read as \(A\) union \(B.\)

Example: Let  \(A = \{ 2,4,6,8\} \) and \(A = \{ 6,8,10,12\} \). Find \(A \cup B\)

We have \(A \cup B = \{ 2,4,6,8,10,12\} \)
\(6\) and \(8\) are the common elements have been taken only once while writing \(A \cup B\).

2. Intersection of Set

The intersection of sets A and B is the set of all elements which are common to both \(A\) and \(B.\) The symbol \(∩\) is used to denote the intersection. The intersection of two sets \(A\) and \(B\) is the set of all those elements which belong to both \(A\) and \(B.\) Symbolically, we write \(A \cap B = \{ X:x \in A\) and \(x \in B\} .\)

Example: Let  \(A = \{ 2,4,6,8\} \) and \(B = \{ 6,8,10,12\} \) Find \(A∩B.\) 

\(6\,\& \,8\) are the only common elements that are common to both \(A\) and \(B\). 
Hence  \(A \cap B = \{ 6,8\} \).

3. Complement of Set

The complement of a set is represented by \(A’,\) is the set of all elements in the given universal set that are not in \(A.\)

Example: Let \(U = \{ 1,2,3,4,5,6,7,8,9,10\} \) and \(A = \{ 1,2,3,4\} \). Find \(A’.\)

From the given sets, we observed that \(1, 6, 7, 9, 10\) are the elements of \(U\) which do not belong to set \(A.\)

So, \(A’ = \{ 1,6,7,9,10\} \).

4. Disjoint Set

If \(B\) and \(C\) are two sets such that \(B ∩ C = ϕ,\) then \(B\) and \(C\) are called disjoint sets.

Example:  let  \(A = \{ 2,4,6,8\} \) and \(B = \{ 1,3,5,7\} \). Then \(A\) and \(B\) are disjoint sets because there are no common elements in set \(A\) and set \(B\).

5. Difference of Sets

The difference of set \(A\) and \(B\) in this order is the set of elements that belongs to \(A\) but not to \(B.\)

Symbolically, we write as \(A-B.\)

Examples: Let  \(A = \{ 2,3,4,6,8\} \) and \(B = \{ 6,4,5,11,14\} \). Find \(A-B.\)

The difference of set \(A\) and \(B\) in this order is the set of elements that belongs to set \(A\) but not to set \(B.\)
Therefore, \(A – B = \{ 2,3,8\} \)

What is the Complement of a Set?

Let \(U\) be the universal set, and \(A\) be a subset of \(U.\) Then, the complement of \(A\) is the set of all elements of \(U\) which are not the elements of \(A.\)

We know that the complement of a set \(A\) can be looked upon, alternatively, as the difference between a universal set \(U\) and the set \(A.\)

Complement of a Set Symbol

Symbolically, we use \(A’\) or \({A^c}\) to represent the complement of set \(A\) with respect to \(U.\) Thus,

\({A^\prime } = \{ x:x \in U\) and \(x \in A\} .\) Obviously \(A’=U–A\)

Universal Set and Complement of a Set

A universal set is a set that contains all the elements or objects of other sets, including its elements. It is typically denoted by \(U\)
The complement of a set A is the difference between a universal set \(U\) and set \(A.\)

Examples of Complement of a Set

(i) Let \(U = \{ 1,2,3,4,5,6,7,8,9,10,11,12\} \) and \(A = \{ 1,3,5,7,9,10\} .\) Find \(A’.\)

 From the given data observed that \(2, 4, 6, 8, 11, 12\) are the elements of \(U\) which do not belong to set \(A.\)

So, \(A’ = \{ 2,4,6,8,11,12\} \).

(ii) Find the complement of \(B\) in \(U.\) If \(B = \{ 1,2,4,6\} \) and \(U = \{ 1,2,4,6,7,8,9\} .\)
By observing the elements, we have \(B’ = \{ 7,8,9\} .\)

Properties of Complement of a Set

1. Complement Laws

The union of a set \(A\) and its complement \(A’\) gives the universal set \(U\) of which \(A\) and \(A’\) are a subset.

\(A \cup {A^\prime } = U\)

The intersection of a set \(A\) and its complement \(A’\) gives the empty set \(∅.\)

\(A∩A’=∅\)

For Example: If \(U = \{ 1,2,3,4,5\} \) and \(A = \{ 1,2,3\} \) then \(A’ = \{ 4,5\} .\)

From the given, we know that,

\(A \cup {A^\prime } = U = \{ 1,2,3,4,5\} \)

Also

\(A∩A’=∅\)

2. Law of Double Complementation

From this law, we see that if we take the complement of the complemented set \(A’\) then, we get the set \(A\) itself.

\((A’)’=A\)

In the previous example we can see that, if \(U = \{ 1,2,3,4,5\} \) and \(A = \{ 1,2,3\} \) then \({A^\prime } = \{ 1,2,3\} \)

Then, the complement of set \(A’.\) we get,

\(\left( {A’} \right)’ = \{ 1,2,3\} = A\)

This gives back set \(A\) itself.

3. Law of Empty Set and Universal Set

From this law, the complement of the universal set gives us the empty set and vice-versa, i.e.,\(∅’=U\) and \(U’=∅\)

Solved Examples – Complement of a Set

Q.1. Find the complement of \(A\) in \(U.\) If \({\rm{A = \{ }}x{\rm{ / }}x\,{\rm{is}}\,{\rm{a}}\,{\rm{number}}\,{\rm{bigger}}\,{\rm{than}}\,{\rm{4}}\,{\rm{and}}\,{\rm{smaller}}\,{\rm{than}}\,{\rm{8\} }}\) and \({\rm{U = \{ }}x{\rm{ / }}x\,{\rm{is}}\,{\rm{a}}\,{\rm{positive}}\,{\rm{number}}\,{\rm{smaler}}\,{\rm{than}}\,7\} .\)
Ans: From the given,
\(A = \{ 5,6,7\} \) and \(U = \{ 1,2,3,4,5,6\} \)
\( \Rightarrow {A^\prime } = \{ 1,2,3,4\} \)
Therefore, the complement of \(A\) is \({A^\prime } = \{ 1,2,3,4\} .\)

Q.2. If \(U = \{ 1,2,3,4,5,6,7\} \) and \(A = \{ 1,2,3,4,5\} \) then find \(A’\)?
Ans: From the given,
\(U = \{ 1,2,3,4,5,6,7\} \) and \(A = \{ 1,2,3,4,5\} \)
\( \Rightarrow A’ = \{ 6,7\} \)
Therefore, the complement of \(A\) is \(A’ = \{ 6,7\} .\)

Q.3 If \(U = \{ 1,2,3,4,5,6,7,8,9,10\} \) and \(A = \{ 1,2,5,6,9\} \). Find A’.
Ans: We have,
\(U = \{ 1,2,3,4,5,6,7,8,9,10\} \) and \(A = \{ 1,2,5,6,9\} \)
So, we get, \(A’ = \{ 2,4,6,8,10\} \)
Therefore, the complement of \(A\) is \(A’ = \{ 2,4,6,8,10\} .\)

Q.4. If \({\rm{U = \{ Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday\} }}\) and
\({\rm{B = \{ Sunday, Monday, Tuesday, Wednesday\} }}{\rm{.}}\) Find \(B’.\)
Ans: We have
\({\rm{U = \{ Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday\} }}\) and
\({\rm{B = \{ Sunday, Monday, Tuesday, Wednesday\} }}{\rm{.}}\)
Then, \({\rm{B’ = \{ Thursday, Friday, Saturday\} }}\)
Therefore, the complement of \(B\) is \(B'{\rm{ = }}{ {\rm{Thursday}},{\rm{Friday}},{\rm{Saturday}}} .\)

Q.5. If \(U = \{ {\rm{All}}\,{\rm{the}}\,{\rm{natural}}\,{\rm{numbers}}\,{\rm{less}}\,{\rm{than}}\,{\rm{or}}\,{\rm{equal}}\,{\rm{to}}\,20\} \) and \(P = \{ {\rm{All}}\,{\rm{the}}\,{\rm{prime}}\,{\rm{numbers}}\,{\rm{less}}\,{\rm{than}}\,{\rm{or}}\,{\rm{equal}}\,{\rm{to}}\,20\} \). Then find \(P’.\)
Ans: From the given,
\(U = \{ {\rm{All}}\,{\rm{the}}\,{\rm{natural}}\,{\rm{numbers}}\,{\rm{less}}\,{\rm{than}}\,{\rm{or}}\,{\rm{equal}}\,{\rm{to}}\,20\} = \{ 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15\} \)
\(P = \{ {\rm{All}}\,{\rm{the}}\,{\rm{prime}}\,{\rm{numbers}}\,{\rm{less}}\,{\rm{than}}\,{\rm{or}}\,{\rm{equal}}\,{\rm{to}}\,20\} = \{ 2,3,5,7,11,13,17,19\} \)
Then, \({P^\prime } = \{ 1,4,6,8,9,10,12,14,15,16,18,20\} \)
Therefore, \({P^\prime } = \{ 1,4,6,8,9,10,12,14,15,16,18,20\} .\)

Q.6 If \(U = \{ m,n,o,p,q,r,s\} \) and \(R = \{ a,m,o,p,r\} \). Find \(R’\).
Ans: \(U = \{ m,n,o,p,q,r,s\} \) and \(R = \{ a,m,o,p,r\} \)
Then, \({R^\prime } = \{ n,q,s\} \)
Therefore, \({R^\prime } = \{ n,q,s\} .\)

Q.7. If \(U = \{ rose,lilly,jasmine,lotus,tulip,iris\} \) and \(F = \{ rose,lilly,iris\} .\) Find \(F’.\)
Ans: \(U = \{ rose,lilly,jasmine,lotus,tulip,iris\} \) and \(F = \{ rose,lilly,iris\} .\)
Then, \({F^\prime } = \{ jasmine,lotus,iris\} \)
Therefore, \({F^\prime } = \{ jasmine,lotus,iris\} .\)

Summary

In this article, we have discussed that sets are representations of well-defined elements and do not vary with perspective. It is typically represented by a letter. Complement of a Set includes all the elements which are not in the given set. We have also discussed types of sets, the difference of two sets, how the difference of two sets helps in finding complements of a set and some of the properties of the complement of a set. At the end, we have discussed some examples for a better understanding of the topic.

Frequently Asked Questions (FAQs) – Complement of a Set

Q.1. What is the Complement of an Empty Set?
Ans: The Complement of an empty set is the Universal Set.

Q.2. What is Complement of a Set?
Ans: The complement of a set is represented by \(A’,\) is the set of all elements in the given universal set that are not in \(A.\)

Q.3. What function has Complement of a Set?
Ans: In set theory, the complement of a set \(A,\) often denoted by \(A’,\) are the elements not in \(A.\) When all sets under consideration to be subsets of a given set \(U,\) the absolute complement of \(A\) is the set of elements in \(U,\) but not in \(A.\)

Q.4. What are examples Complement of a Set?
Ans: Examples of Complement of a Set are,
If \(U = \{ 1,2,3,4,5,6,7,8,9\} \) and \(A = \{ 1,2,3,4,5\} \) then \(A’\) is,
\({A^\prime } = \{ 6,7,8,9\} \)
If \(U = \{ a,b,c,d,e,f,g,h\} \) and \(B = \{ c,d,e,f\} \) then \(B’\) is,
\({B^\prime } = \{ a,b,g,h\} .\)

Q.5. What is the symbol for Complement of a Set?
Ans: Symbolically, we use \(A’\) or \({A^C}\) to represent the complement of set \(A\) with respect to \(U.\)