Difference of Set: Definition, Formula, Properties, Examples

What is the Difference of Sets in Set Theory?

In set theory and discrete mathematics, the difference of sets is one of the most important set operations used to remove or exclude elements from a group. While union combines elements and intersection finds common elements, the set difference operation focuses on identifying elements that belong to one set but not to another.

Difference of Sets: Definition

The difference of two sets $A$ and $B$ (in that order) refers to the set of elements that are present in $A$ but not in $B$. In simple words, we subtract the elements of $B$ from $A$.

Symbolically, this is written as $A - B$, which is read as “A minus B”.

Difference of Sets: Notation in Mathematics

Mathematically, the set difference is defined as $A - B = {x : x \in A \text{ and } x \notin B}$, meaning the result contains only those elements that belong to $A$ and not to $B$.

Difference of Sets: Examples with Solutions

Practising examples helps you clearly understand how set subtraction works in real problems.

Example 1: Numerical Sets

Let $A = \{1, 2, 3, 4\}$ and $B = \{4, 5, 6, 8\}$.

Then $A - B = \{1, 2, 3\}$, which are elements in $A$ but not in $B$, and $B - A = \{5, 6, 8\}$, which are elements in $B$ but not in $A$.

Example 2: Real-Life Categories

Let $A = \{\text{blue, violet, green, yellow, black, white, red}\}$ and $B = \{\text{black, blue, red, grey, brown}\}$.

Then $A - B = \{\text{violet, green, yellow, white}\}$ and $B - A = \{\text{grey, brown}\}$.

This example shows how the difference operation removes shared items.

Example 3: Set-Builder Form

Let $A = \{x : x \in \mathbb{N} \text{ and } x \leq 9\}$ and $B$ be the set of integers greater than $-5$ and less than or equal to $5$.

So $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ and $B = \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}$.

Then $A - B = \{6, 7, 8, 9\}$ and $B - A = \{-4, -3, -2, -1, 0\}$.

Example 4: Equal Sets

Let $A$ be the set of all even natural numbers less than $10$ and $B = \{2, 4, 6, 8\}$.

Since both sets are equal, $A - B = \phi$ and $B - A = \phi$.

This shows that when two sets are identical, their difference is always the empty set.

These difference of sets examples clearly demonstrate how to subtract elements from one set based on their presence in another set. In every case, only the non-common or unique elements remain in the final result.

How to Find the Difference of Sets Step-by-Step

To find the difference between two sets, follow this systematic method:

Step 1: Write the given sets clearly in roster or set-builder form.

Step 2: Check the order carefully, because $A - B$ and $B - A$ give different results.

Step 3: Find the common elements using the intersection $A \cap B$.

Step 4: Remove these common elements from the first set.

Step 5: The remaining elements form the difference set. Hence, $A - B$ contains only elements that belong to $A$ but not to $B$.

Difference of Sets: Venn Diagram

The difference of sets Venn diagram visually represents the elements that belong to one set but not to another. When we write $A - B$, it means we are looking for all the elements in Set $A$ that are not present in Set $B$.

This concept is best understood with the help of a Venn diagram, where overlapping areas show common elements, and the shaded part represents the set difference.

Example Using Venn Diagram:
Let $A = \{1, 2, 3, 4\}$ and $B = \{4, 5, 6, 8\}$.

From the Venn diagram:

• The overlapping part (i.e., $A \cap B$) is $\{4\}$
• The difference $A - B$ includes elements in $A$ but not in $B$, which is $\{1, 2, 3\}$
• Similarly, $B - A$ includes elements in $B$ but not in $A$, which is $\{5, 6, 8\}$

Thus, the Venn diagram for the difference of sets clearly shows how the common element is excluded from the result, helping learners visually understand the subtraction between sets.

Properties of the Difference of Sets

Understanding the properties of difference of sets in mathematics helps clarify how this operation behaves under different conditions. These properties are useful in solving problems related to set operations, Venn diagrams, and logical reasoning.

Here are the key properties of set difference:

1. Non-Commutative Property:
   In general, $A - B \ne B - A$.
   This means that set difference is not commutative, and the order of subtraction matters.
2. Identity Property with Same Set:
   The difference of a set with itself is always the empty set:
   $A - A = \phi$ or $\phi$
3. Difference with Empty Set:
   The difference of any finite set with the null set gives the original set:
   $A - \phi = A$
4. Difference with Universal Set:
   The difference of a set with the universal set results in the empty set:
   $A - U = \phi$ (Since all elements of $A$ are already included in $U$)
5. Mutually Disjoint Results:
   The sets $A - B$, $A \cap B$, and $B - A$ are all mutually disjoint, meaning:
   • $(A - B) \cap (A \cap B) = \phi$
   • $(A - B) \cap (B - A) = \phi$
   • $(A \cap B) \cap (B - A) = \phi$

Symmetric Difference of Sets $(A \Delta B)$

The symmetric difference of sets refers to the set of elements that belong to either set $A$ or set $B$ but not to both. In other words, it excludes the common elements.

The symmetric difference of two sets $A$ and $B$ is defined as:
$A \Delta B = (A - B) \cup (B - A)$
Alternatively, it can also be represented as:
$A \Delta B = (A \cup B) - (A \cap B)$

This operation is useful in identifying non-overlapping elements between sets and is commonly applied in data filtering, logic circuits, and set-based problem-solving.

How to Find Symmetric Difference of Sets

To find the symmetric difference between two sets, follow these simple steps:

1. Identify the given non-empty sets and express them in set-builder or roster form.
2. Find the difference $A - B$ (elements in $A$ but not in $B$).
3. Find the difference $B - A$ (elements in $B$ but not in $A$).
4. Take the union of both differences: $A \Delta B = (A - B) \cup (B - A)$

Symmetric Difference of Sets – Examples

Example 1:
Let $A = \{1, 2, 3, 4\}$ and $B = \{4, 5, 6, 8\}$
Then:
$A - B = \{1, 2, 3\}$,
$B - A = \{5, 6, 8\}$
So, $A \Delta B = \{1, 2, 3, 5, 6, 8\}$

Example 2:
Let $A = \{\text{blue, violet, green, yellow, black, white, red}\}$
and $B = \{\text{black, blue, red, grey, brown}\}$
Then:
$A - B = \{\text{violet, green, yellow, white}\}$
$B - A = \{\text{grey, brown}\}$
So, $A \Delta B = \{\text{violet, green, yellow, white, grey, brown}\}$

Example 3:
Let $A = \{x \in \mathbb{N} \mid x \leq 10\}$
Let $B$ = set of all integers greater than $-5$ and less than or equal to $5$

Now,
$A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$
$B = \{-4, -3, -2, -1, 0, 1, 2, 3, 4, 5\}$
Then:
$A - B = \{6, 7, 8, 9\}$
$B - A = \{-4, -3, -2, -1, 0\}$
So, $A \Delta B = \{6, 7, 8, 9, -4, -3, -2, -1, 0\}$

Example 4:
Let $A$ = set of all even natural numbers less than $10$
Let $B = {2, 4, 6, 8}$
Then:
$A = {2, 4, 6, 8}$, $B = {2, 4, 6, 8}$
$A - B = \phi$, $B - A = \phi$
So, $A \Delta B = \phi$

Symmetric Difference of Sets – Venn Diagram

The symmetric difference of sets Venn diagram shows the regions of both sets $A$ and $B$ excluding their intersection. The shaded areas represent elements that are either in $A$ or in $B$, but not in both.

For instance, let $A = \{1, 2, 3, 4\}$ and $B = \{4, 5, 6, 8\}$
Then:

• $A \cap B = \{4\}$ (common element)
• $A \Delta B = \{1, 2, 3, 5, 6, 8\}$ (all non-common elements)

The Venn diagram will shade only the non-overlapping parts of the two circles, clearly illustrating the symmetric difference.

Relationship Between Difference, Union, and Intersection

In set theory and discrete mathematics, the three most important set operations are union, intersection, and difference. These operations work together to help us combine sets, find common elements, or remove unwanted elements.

Think of them like three tools:

• Union → combines everything

• Intersection → keeps only common parts

• Difference → removes common parts

Understanding the relationship between union, intersection, and difference of sets is essential for solving Venn diagram questions, probability problems, logical reasoning, and data classification tasks.

Important Formulae for Difference and Symmetric Difference of Sets

In set theory and discrete mathematics, the difference of sets removes common elements, while the symmetric difference keeps only the non-common elements from both sets. These operations are closely connected to union and intersection, so learning their formulas helps you simplify problems faster.

Here’s a clear summary table.

Difference and Symmetric Difference Formula Table

Symmetric Difference Formulae

Solved Examples Based On the Difference of Sets

Example 1: If $\mathbf{A}, \mathbf{B}$, and $\mathbf{C}$ are non-empty sets, then $(A-B) \cup(B-A)$
1) $(A \cup B)-B$
2) $A-(A \cap B)$
3) $(A \cup B)-(A \cap B)$
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: If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are any three sets,then $A-(B \cup C)$ is equal to?
Option 1) $(A-B) \cup(A-C)$
Option 2) $(A-B) \cap(A-C)$
Option 3) $(A-B) \cup C$
Option 4) $(A-B) \cap C$

Solution:
The difference of the sets $A$ and $B$ in this order is the set of elements which belong to $A$ but not to $B$ wherein Symbolically, we write A - B and read as "A minus B".
According to De Morgan's laws
$A-(B \cup C)=(A-B) \cap(A-C)$
Hence the answer is option 2.

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:

$ \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 $A \subset B$, then $A-B=$
1) A
2) B
3) $A \cap B$
4) $\phi$

Solution:
If A is subset of B , then Venn diagram looks something like this:

Clearly, $\mathrm{A}-\mathrm{B}=\phi$, as all elements of A are present in B.

List of Topics Related to the Difference of Sets

To fully understand the difference of sets, it's important to explore related concepts in set theory. This section lists key topics that build the foundation for mastering set operations and their applications.

NCERT Resources

Explore essential NCERT study resources for Class 11 Sets, including detailed solutions, clear revision notes, and carefully selected exemplar problems. These materials are designed to strengthen your understanding and help you prepare confidently for board exams and competitive tests.

NCERT Solutions for Class 11 Chapter 1 Sets

NCERT Notes for Class 11 Chapter 1 Sets

NCERT Exemplar for Class 11 Chapter 1 Sets

Practice Questions based on Difference of Sets

Regular practice is key to mastering the difference of sets and its related operations. Below, you'll find a set of practice questions designed to test your understanding of definitions, formulas, properties, and real-world applications of set difference. Use these exercises to build accuracy and confidence.

Practice Here: Difference Of Set - Practice Question MCQ

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