Difference of Set: Definition, Formula, Properties, Examples

The difference between the two collections is a concept that examines the presence of elements in one collection but not in another. For instance, consider two groups of employees in a company: In one group, some members participated in a mandatory training session, but the other group did not participate in it at all. Therefore, the possibilities in the second set that are not in the first set indicate the number of employees who have not experienced the training. This operation is essential for defining the specific problem and issue domains or domains that require a solution in specific sectors or segments within a broad classification.

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In this article, we will cover the concept of the difference of sets. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of one question has been asked on this concept, including one in 2021.

Set

Sets are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V…...

Difference of sets
The difference of the sets $A$ and $B$ in this order is the set of elements that belong to $A$ but not to B.

Symbolically, we write A - B and read as "A minus B".
For example, If $A=\{1,2,3,4\}$ and $B=\{4,5,6,8\}$,
Then, $A-B=\{1,2,3\}$ and $B-A=\{5,6,8\}$
U
(A-B)

The sets $A-B, A \cap B$ and $B-A$ are mutually disjoint sets, i.e., the intersection of any two of these sets is the null set as shown in figure.

How to find the difference of sets
Following are the steps to find the difference of sets:-
- Identify the given non-empty sets and write them in set-builder form.
- Identify the order of difference, i.e., if we are asked to find A - B or B - A.
- Find the intersection of the sets $A$ and $B$.
- Eliminate the elements of $A \cap B$ from Set $A$ to write a new set. In other words, remove the elements in $A$ that are also present in Set $B$.
- Define the new set as $A-B . A-B$ is the set of elements of $A$ that are not in $B$.

Properties of Difference of Sets

1. In general A - B does not equal B - A. This means that the difference of sets are not interchangeable.

2. When we take the difference of the same set then the result is empty.

$\mathrm{A}-\mathrm{A}=\phi$

3. When we take the difference of a finite set with the null set then the result is a finite set.

$\mathrm{A}-\phi=\mathrm{A}$

4. When we take the difference of a finite set with a universal set then the result is empty.

$A-U=\phi$

Note:

If A is a subset of B, then $\mathrm{A}-\mathrm{B}=\phi$

Symmetric Difference of Sets (A B )
The symmetric difference of two sets $A$ and $B$ is defined as

$
A \Delta B=(A-B) \cup(B-A)
$

Venn Diagram

Clearly, $A \Delta B$ also equals $(A \cup B)-(A \cap B)$

Summary :Based on this perspective, the difference of the sets is considered to be one of the most powerful instruments within set theory because it possible to indicate some constituents being contained within one set but absent in another set. This is very helpful in real-life circumstances such as identifying which among the number of workers has not completed a given course at the workplace. It is reliable to designate the activities based on the difference of sets and how it will be used to assign the activities, and resources and to identify specific drawbacks in larger populations.

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Solved Examples Based On the Difference of Sets:

Example 1: If $\mathbf{A}, \mathbf{B}$, and $\mathbf{C}$ are non-empty sets, then $(A \cup B)-(A \cap B)$
1) $(A \cup B)-B$
2) $A-(A \cap B)$
3) $(A-B) \cup(B-A)$
4) $(A \cap B) \cup(A \cup B)$

Solution:

Clearly, as the sets in the question and in the third option, both equal the symmetric difference of $A$ and $B$, so both these are equal.

$
(A-B) \cup(B-A)=(A \cup B)-(A \cap B)
$

Hence, the answer is the option 3.
Example 2:

$
\begin{aligned}
& \mathrm{A}=\left\{\mathrm{n} \in \mathrm{N} \mid \mathrm{n}^2 \leq \mathrm{n}+10,000\right\}, \mathrm{B}=\{3 \mathrm{k}+1 \mid \mathrm{k} \in \mathrm{N}\} \\
& C=\{2 \mathrm{k} \mid \mathrm{k} \in \mathrm{N}\} A \cap(B-C)
\end{aligned}
$

$ \text { Let } \mathrm{A}=\left\{\mathrm{n} \in \mathrm{N} \mid \mathrm{n}^2 \leq \mathrm{n}+10,000\right\}, \mathrm{B}=\{3 \mathrm{k}+1 \mid \mathrm{k} \in \mathrm{N}\} \text { and } C=\{2 \mathrm{k} \mid \mathrm{k} \in \mathrm{N}\} \text {, then the sum of all the elements of the set } A \cap(B-C) \text { is equal to } $

Solution:

$\begin{aligned} & A: n^2-n \leq 10,000 \\ & \Rightarrow n(n-1) \leq 100 \cdot 100 \\ & \Rightarrow n=\{1,2, \ldots, 100\} \\ & B=\{4,7,10,13, \ldots\} \\ & C=\{2,4,6,8,10, \ldots\} \\ & B-C=\{7,13,19, \ldots\} \\ & \begin{aligned} A \cap(B-C)=\{7,13,19, \ldots, 97\} \\ \text { Sum }=\frac{16}{2}\{14+14.6\} \\ \quad=832\end{aligned}\end{aligned}$

Hence, the answer is 832.

Example 3:

Let $\mathrm{A}, \mathrm{B}$, and $\mathbf{C}$ be sets such that $\phi \neq A \cap B \subseteq C$. Then which of the following statements is not true?
1) $B \cap C \neq \phi$
2) If $(A-B) \subseteq C$, then $A \subseteq C$
3) $(C \cup A) \cap(C \cup B)=C$
4) If $(A-C) \subseteq B$, then $A \subseteq B$

Solution:

DIFFERENCE OF SETS -
The difference of the sets $A$ and $B$ in this order is the set of elements that belong to $A$ but not to B.
- wherein

Symbolically, we write A - B and read as "A minus B".

$
\begin{aligned}
& \text { As }(A \cap B) \subseteq C \\
& \Rightarrow(A \cap B) \subseteq(B \cap C) \\
& \text { as }(A \cap B) \neq \phi_{\Rightarrow}(B \cap C) \neq \phi
\end{aligned}
$
So, option (1) is true

Let $x \in A$ and $x \notin B \Rightarrow x \epsilon(A-B)_{\Rightarrow} \Rightarrow x \epsilon C$ let $x \in A$ and $x \epsilon B \Rightarrow x \epsilon(A \cap B) \Rightarrow x \epsilon C$ Hence $x \in A$ and $x \in C \Rightarrow A \subseteq C$ So, option (2) is true.

Let

$\begin{aligned} & x \in C, x \in(C \cup A) \cap(C \cup B)) \\ & =>x \epsilon(C \cup A) \text { and } x \epsilon(C \cup B)) \\ & (x \in C \text { or } x \in A) \text { and }(x \in C \text { or } x \in B) \\ & =>x \epsilon C \text { or } x \epsilon(A \cap B) \\ & =>x \epsilon C \quad A s, A \cup B \subseteq C\end{aligned}$

$\begin{aligned}
&(C \cup A) \cap(C \cup B) \subseteq C.......\text { (1) }
\end{aligned}$

Now,
$
\begin{aligned}
& =>x \epsilon C \\
& =>x \epsilon(C \cup A) \cap(C \cup B) \\
& =>C \subseteq(C \cup A) \cap(C \cup B) ......(2)
\end{aligned}
$

From (1) and (2)

$
(C \cup A) \cap(C \cup B)=C
$

=> Option (3) is correct.
For $\mathrm{A}=\mathrm{c}, A-C=\phi$

$
=>\phi \subseteq B \text { but }=>A \nsubseteq B
$
So, option (4) is not true.

Example 4:

If $A=(x \in \mathbf{R}:|x|<2)$ and $B=(x \in \mathbf{R}:|x-2| \geqslant 3)$; then:
1) $A-B=[-1,2)$
2) $B-A=\mathbf{R}-(-2,5)$
3) $A \cup B=\mathbf{R}-(2,5)$
4) $A \cap B=(-2,-1)$

Solution:

$\begin{aligned} & A=\{x: x \in(-2,2)\} \\ & B=\{x: x \in(-\infty,-1] \cup[5, \infty)\} \\ & A \cap B=\{x: x \in(-2,-1]\} \\ & A \cup B=\{x: x \in(-\infty, 2) \cup[5, \infty)\} \\ & A-B=\{x: x \in(-1,2)\} \\ & B-A=\{x: x \in(-\infty,-2] \cup[5, \infty)\}\end{aligned}$

Hence, the answer is the option 2.

Frequently Asked Questions(FAQ)-

1. What is the difference of sets?

Ans: The difference of sets $A$ and $B$ in this order is the set of elements that belong to $A$ but not to B . $\square$
2. What are disjoint sets?

Ans: The intersection of any two of these sets is the null set is called a disjoint set.
3. What is the symmetric difference of sets?

Ans: The symmetric difference of two sets $A$ and $B$ is defined as
$A \Delta B=(A-B) \cup(B-A)$
4. What is the value of $A-A$ ?

Ans: \$imathrm $\{\mathrm{A}\}$-Imathrm $\{\mathrm{A}\}=$ \phi\$
5. Is A-B equal to B-A?

Ans: No