Difference of Sets

A set is a collection of elements. These elements can be numbers, names, letters, or any clearly defined objects. A set is written using curly brackets { }.
 

The difference of sets is an operation in set theory that identifies elements present in one set but not in another. If you have two sets A and B, the difference between A and B includes the elements that are only in A. It is written as A – B = {x ∈ A | x ∉ B}.
 

Both set intersection and set difference are operations in set theory, but they serve different purposes.
 

The difference of sets helps identify elements that are unique to a set when compared to another set. Understanding the properties of the difference of sets helps us easily solve problems related to the difference of sets. 

• If two sets A and B have the same elements, then their difference is the empty set. In other words:
  A – B = ∅ and B – A = ∅
  
   
• When we subtract the empty set from any set, we’re not removing anything because the empty set has no elements. So there is no change to set A, and the result is just: A – ∅ = A.
  
   
• If the intersection of two sets is empty, it means the sets have no common elements, and the difference between them is just the first set. 
  In symbols: If A ∩ B = ∅, then A – B = A.
  
   
• If set A is a subset of set B, then subtracting B from A gives the empty set. This is because all elements of A are already in B, so there's nothing left after subtraction. In symbols: 
  If A  B, then A – B = ∅ 

To find the difference of two sets, identify the elements that belong to the first set but not to the second. In symbols:
A – B = {x ∈ A |x ∉ B}

Steps to find the difference of sets A – B

• List all the elements of sets A and B
• Identify which elements in A are also found in B 
• Remove those common elements from set A
• The remaining elements form A – B

For example,
A = {1, 2, 3, 4, 5}  B = {4, 5, 6, 7, 8, 9}

List all the elements from sets A and B, but don’t repeat any elements. So the elements are 1, 2, 3, 4, 5, 6, 7, 8, 9.

Check the elements that are also in set B, which are 4 and 5

Remove from the set of A 

So, A – B = {1, 2, 3}.
 

Order matters in the difference of sets — just like 7 - 5  5 - 7, the difference A - B is not the same as B - A. Reversing the order of the sets in a different operation can lead to a completely different result. 
For example, 
A = {1, 2, 3, 4, 5} B = {4, 5, 6, 7, 8, 9, 0}
A – B = {1, 2, 3, 4, 5} - {4, 5, 6, 7, 8, 9, 0}
A – B = {1, 2, 3}

B – A = {4, 5, 6, 7, 8, 9, 0} - {1, 2, 3, 4, 5}
B – A = {6, 7, 8, 9, 0}

A – B ≠ B – A
 

A Venn diagram is used to represent relationships between two sets visually. The difference of sets is written as A – B, which means the elements in set A are not in set B.

The image shows that sets A and B are overlapping. 

Difference of Three Sets

The difference of three sets refers to the elements that are in set A, but not in set B or set C. The difference of the three sets is written as A – B – C = {x  A | x  B and x  C}. 

Symmetric Difference of Sets

The symmetric difference of sets A and B includes the elements that are in     A or in B, but not in both. In other words, the symmetric difference includes only the elements that are unique to each set. The symmetric difference of sets is defined as A Δ B = {x| x ∈ A |x ∈ B, but x∉ A∩B}.

Relation Between Set Complement and Set Difference

The complement of a set is closely related to the concept of set difference. It represents all the elements in the universal set that are not in the given set. In other words, the complement of set P is the difference between the universal set U and P.

This relationship is written as: 

P′ = U – P

The concept of difference of sets has many real-life applications. Some of them are mentioned below:  

• Attendance: Teachers can use set differences to find absent students. If A is the full class list and B is the list of students present, then A - B shows who was absent.

• Customer Promotions: Companies use set differences to target customers who haven't received an offer. If A is all customers and B is those who already got the offer, then A - B gives the list for new promotions.

• Invitation List: To invite only new people, subtract last year’s invitees (B) from this year’s contacts (A). Then A - B gives the new invitees.

• Engineering: Engineers track unused materials by subtracting used items (B) from the full inventory (A). So A - B shows what's still available.  

• Video Editing: Editors identify unused clips by subtracting the clips in the final video (B) from all recorded clips (A). So A - B lists the unused footage.