Difference of Sets: Definition, Formula, Examples

A well-defined collection of objects or elements is known as a set. Any set consisting of all the things or elements related to a particular context is defined as a universal set. The difference of sets \(A\) and \(B\) is the set of elements that belongs to set \(A\) but not to set \(B.\). Therefore, \(A – B = \left\{{2,3,8} \right\}\)

Sets

A set is a well-defined collection of objects. All the objects in a set should have a common feature or property, and it should be possible to infer whether any given object belongs to the set or not.

Let us understand sets through some examples.

• 1. The collection of all the boys in a class represents a set.
• 2. The number of books in a school library represents a set.

Types of Sets

Sets are of the following types:

• 1. Singleton set
• 2. Empty set
• 3. Finite set
• 4. Infinite set
• 5. Equal set

Singleton Set

A set that contains only one element is called a Singleton Set.

Example: \(A = \){\(x:x\) is neither prime nor composite natural number}

Empty Set

A set that does not contain any element is called an Empty set, or a Null set, or a Void set. An empty set is denoted by the symbol \(\varphi \) or \(\{ \,\} .\)

Example:\(P = \){ \(x:x\) is a leap year between \(2017\) and \(2019\)}

Finite Set

Consider the following sets:

\(A = \) {the students of your school}
\(L = \left\{{P,Q,R,S} \right\}\)

In set \(A\) the number of elements is the number of students in your school. And, in set \(L,\) the number of elements is \(4.\) Since it is possible to count the number of elements in both sets \(A\) and \(L,\) i.e. they contain finite elements, such sets are called finite sets.

Infinite Sets

Consider the following sets:

\(B = \){\(x:x\) is an even number}
\(J = \) {\(x:x\) is a multiple of \(7\)}

Since set \(B\) contains all the even numbers that cannot be counted, the number of elements of this set is not finite.

Similarly, all the elements of \(J\) cannot be listed. We observed that the number of elements in \(B\) and \(J\) is infinite. Hence, such sets are called infinite sets.

Equal Sets

Consider the following sets:

\(A = \) {Sachin,Dravid,Kohli}
\(B = \) {Dravid,Sachin,Dhoni}
\(C = \) {Kohli,Dravid,Sachin}

All the names that are in set \(A\) are in set \(C\) but not in set \(B.\) Thus, \(A\) and \(C\) have the same elements, but some elements of \(A\) and \(B\) are different. So, sets \(A\) and \(C\) are equal sets but sets \(A\) and \(B\) are not equal.

Two sets \(A\) and \(C\) are equal if every element in \(A\) belongs to \(C\) and every element in \(C\) belongs to \(A.\) Two equal sets, \(A\) and \(C,\) are represented as \(A = C.\)

Basic Operations on Sets

The basic operations on sets are:

Union of Sets

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

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

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

Intersection of Sets

The intersection of sets \(A\) and \(B\) is the set of all elements which are common to both \(A\) and \(B.\)

The symbol \( \cap \) is used to denote the intersection. The intersection of sets \(A\) and \(B\) is the set of all those elements which belong to both \(A\) and \(B.\) Symbolically, the intersection is represented as \(A \cap B = \) {\(x:x \in A\) in and \(x:x \in B.\)}

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

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

Complement of Sets

The complement of sets is the set of all elements in the given universal set that are not in \(A.\) It is represented as \(A’.\)

Example: Let \(U = \left\{{1,2,3,4,5,6,7,8,9,10} \right\}\) and \(A = \left\{{1,2,4,5} \right\}.\) Find \(A’.\) In the above example, it can be observed that \(3,6,7,8,9,10\) are the elements of \(U\) which do not belong to set \(A.\) So, \(A’ = \left\{{3,6,7,8,9,10} \right\}.\)

Disjoint in Sets

In the below picture, \(B\) and \(C\) are two sets such that \(B \cap C = \phi ,\) then \(B\) and \(C\) are called disjoint sets.

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

Define Difference of Sets

The difference of sets \(A\) and \(B\) is the set of elements that belongs to \(A\) but not to \(B.\) Symbolically, the difference of sets is represented as \(A – B.D\)

We denote the difference of \(A\) and \(B\) by \(A – B\) or simply \(A\) minus \(B\) and the difference of \(B\) and \(A\) by \(B – A\) or simply \(B\) minus \(A.\)

So, \(A – B = \) {\(x:x \in A\) and \(x \in B\)} and \(B – A = \){\(x:x \in B\) and \(x \in A\)}

Difference of Sets Example

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

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

Difference of Sets Facts

• (i) If set A and set B are equal then, \(A – B = A – A = \phi \) (empty set).
• (ii) If an empty set is subtracted from a set, then the difference is given set itself, i.e., \(A – \phi = A.\)
• (iii) If a set is subtracted from an empty set, then the result is an empty set, i.e., \(\phi – A = \phi .\)
• (iv) If a superset is subtracted from a subset, then the difference is an empty set, i.e., \(A – B = \phi \) if \(A \subset B.\)
• (v) If \(A\) and \(B\) are disjoint sets then, \(A – B = A\) and \(B – A = B.\)

Readers can check Difference of Sets Example and practise problems below.

Difference of Sets Practices Problems With Solutions

Q.1. \(A = \left\{{2,4,7,9,11,12} \right\}\) and \(B = \left\{{1,2,3,4,5} \right\}.\) Find \(A – B\) and \(B – A.\)
Ans: We can say that \(A – B = \left\{{7,9,11,12} \right\}\) as these elements belong to \(A\) but not to \(B.\) And, \(B – A = \left\{{1,3,5} \right\}\) as these elements belong to \(B\) but not to \(A.\)

Q.2. \(A = \left\{{a,b,c,d,e,f,g,h} \right\}\) and \(B = \left\{{a,e,f,g,h,i,j} \right\}.\) Find \(A – B\) and \(B – A.\)
Ans: The unique elements in \(A\) are \(b,c,d,\) and the unique elements in \(B\) are \(I\) and \(j.\)
Thus, \(A – B = \left\{{b,c,d} \right\}\) and \(B – A = \left\{{i,j} \right\}\)

Q.3. If \(A = \left\{{1,2,6,8,9} \right\}\) and \(A \cap B = \left({1,8} \right).\) Find \(A – B.\)
Ans: The intersection of \(A\) and \(B,\) \(\left({A \cap B} \right)\) is the set of all elements common in both \(A\) and \(B.\)
Therefore, \(\left({A – B} \right) = A – \left({A \cap B} \right)\)
\( \Rightarrow \left({A – B} \right) = \left\{{2,6,9} \right\}\)
Therefore, \( \Rightarrow \left({A – B} \right) = \left\{{2,6,9} \right\}\)

Q.4. \(M = \left\{{a,e,i,o,u} \right\}\) and \(N = \left\{{a,e,k,u} \right\}.\) Find \(M – N.\)
Ans: The elements in only \(M\) are \(i,o.\)
Therefore, \(\left({M – N} \right) = \left\{{i,o} \right\}.\)

Q.5. \(X = \left({a,b,c,d,e} \right)\) and \(Y = \left({f,b,d,g} \right).\) Find \(X – Y.\)
Ans: The intersection of \(X\) and \(Y\) \(\left({X \cap Y} \right)\) is the set of all elements common in both \(X\) and \(Y.\)
Therefore, \(\left({X – Y} \right) = X – \left({X \cap Y} \right).\)
\( \Rightarrow \left({X – Y} \right) = \left\{{a,b,c,d,e} \right\} – \left\{{b,d}\right\}\)
\( \Rightarrow \left({X – Y} \right) = \left\{{a,c,e} \right\}\)
Therefore, \(\left({X – Y} \right) = \left\{{a,c,e} \right\}.\)

Q.6. Given the sets \(X = \left\{{a,b,c,d,e,f,g,h} \right\}\) and \(Y = \left\{{f,b,d,g,h} \right\}.\) Find the value of the set \(X – Y.\)
Ans: The intersection of \(X\) and \(Y,\) \(\left({X \cap Y} \right)\) is the set of all elements common in both \(X\) and \({Y.}\)
Therefore, \(\left({X – Y} \right) = X – \left({X \cap Y} \right).\)
\( \Rightarrow \left({X – Y} \right) = \left\{{a,b,c,d,em,f,g,h} \right\} – \left\{{b,d,f,g,h} \right\}\)
\( \Rightarrow \left({X – Y} \right) = \left\{ {a,c,e} \right\}\)
Therefore, \(\Rightarrow \left({X – Y} \right) = \left\{{a,c,e} \right\}.\)

Summary

The Difference of Sets \(A\) and \(B\) in this order is the set of elements that belongs to set \(A\) but not to set \(B.\) We denote the difference of \(A\) and \(B\) by \(A – B\) or simply \(A\) Minus \(B.\) And the difference of \(B\) and \(A\) by \(B – A\) or simply \(B\) minus \(A.\) The properties of sets are union, intersection, complement, and difference.

FAQs on Difference of Sets

Q.1. How do you find the Difference of a Set?
Ans: To find the difference of sets \(A\& B,\) we begin by writing all the elements of A and then remove the elements of \(A\) that is also an element of \(B.\)

Q.2. What are the types of Sets?
Ans: There are different types of Sets are,
1. Empty Set
2. Singleton Set
3. Finite Set
4. Infinite Set
5. Equal Set
6. Universal Set
7. Subset

Q.3. Explain the Difference of Sets with example?
Ans: The difference of sets \(A\) and \(B\) in this order is the set of elements that belongs to \(A\) but not to \(B.\)
Examples: Let \(A = \left\{{1,2,3,4,5,6,8} \right\}\) and \(B = \left\{{3,4,5,12,14} \right\}.\) Find \(A – B.\)
So, here the set of elements that belongs to set \(A\) but not to set \(B.\)
Therefore, \(A – B = \left\{{1,2,6,8} \right\}\)

Q.4. What does ∩ mean in sets?
Ans: The symbol \( \cap \) is used to denote the intersection. The intersection of sets, \(A\) and \(B\) is the set of all those elements that belong to \(A\) and \(B,\) represented as \(A \cap B.\)

Q.5. What is \(A\) minus \(B\) in sets?
Ans: \(A\) minus \(B\) in sets means the set of elements that belongs to \(A\) but not to \(B.\)

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