Difference of Sets

Difference of Sets is the operation defined on sets, just like we can perform arithmetic operations on numbers in mathematics. Other than the difference, we can also perform the union and intersection of sets for any given set. These operations have a lot of important applications in mathematical practice.

In this article, we will learn about the difference of sets, including its definition, mathematical expressions, Venn diagram, as well as properties of the difference of sets.

What is a Set?

In mathematics, a set is a collection or grouping of well-defined objects. All such objects, when grouped together in a set, are called elements. Sets are represented by capital letter symbols, and the elements are placed together in a curly bracket {}.

For example, if W is the set of whole numbers, then W = {0, 1, 2, 3, 4, 5,....,∞}.

Learn more about Sets and their types

Difference of Two Sets

The difference of the two sets P and Q is nothing but another set, say R, which consists of all the elements present in the first set and not the second.

Here, the first and second sets mean the order of the sets around the subtraction sign. In other words, whenever the difference between two sets P and Q is to be found, it is done so by removing the common elements, i.e., elements of P ∩ Q from set P. Similarly, whenever the difference between Q and P is to be found, it is done by removing common elements of P ∩ Q from set Q.

Mathematical Definition of Difference of Sets

The difference of two sets, P and Q, in this order, is defined as the set of all those elements that are present in set P but that are not present in set Q.  This can be mathematically expressed as

P – Q = {x / x ∈ P and x ∉ Q}: removing elements of P ∩ Q from set P.
OR
Q – P = {x / x ∈ Q and x ∉ P}: removing elements of P ∩ Q from set Q.

How to Find the Difference of Sets?

The difference between sets can be found with the help of the following steps:

• Step 1: Identify the given non-empty sets and write them in set-builder form.
• Step 2: Identify the order of difference, i.e., if we are asked to find P – Q or Q – P.
• Step 3: Express the difference in mathematical form.
• Step 4: Strike off all the common elements present in both the given sets.
• Step 5: All elements left in the first set after removing common elements are the difference of the two sets.

Difference of Sets Example

Let's consider an example of the difference of sets for better understanding.  

Example: If P = {5, 10, 15, 20, 25, 30} and Q = {10, 20, 30, 40, 50, 60}, then find:

• P – Q
• Q – P

Solution:

P – Q implies all elements present in P and not in Q.

Step 1. Express the difference in mathematical form.
P – Q = {5, 10, 15, 20, 25, 30} – {10, 20, 30, 40, 50, 60}

Step 2. Strike off all the elements present in both P and Q.
{5, 10, 15, 20, 25, 30} – {10, 20, 30, 40, 50, 60}.
P – Q is the set of elements left in set P afterwards, i.e.,
P – Q = {5, 15, 25}.

Now, Q – P implies all elements present in Q and not in P.

Step 1. Express the difference in mathematical form.
Q – P = {10, 20, 30, 40, 50, 60} – {5, 10, 15, 20, 25, 30}.

Step 2. Strike off all the elements present in both P and Q.
{10, 20, 30, 40, 50, 60} – {5, 10, 15, 20, 25, 30}.
Q – P is the set of elements left in set Q afterwards, i.e.,
Q – P = {40, 50, 60}.

Order of Difference

While finding the difference of two sets, it is very important to keep the order of the difference in mind. Just like the subtraction of two numbers is not commutative (9 – 0 ≠ 0 – 9), the difference of any two sets is not commutative, i.e., P – Q ≠ Q – P. This means that changing the order of the sets while subtracting may alter the results.

Example: If P = {4, 5, 6, 7, 8} and Q = {6, 7, 8, 10}, is P – Q = Q – P?

Solution:

LHS = P – Q = {4, 5, 6, 7, 8} - {6, 7, 8, 10} = {4, 5}
RHS = Q – P = {6, 7, 8, 10} - {4, 5, 6, 7, 8} = {10}
LHS ≠ RHS

Thus, P – Q ≠ Q – P

Venn Diagram of Difference of Sets

In the Venn diagram below, the pink-shaded region depicts the set P – Q, where all the elements of set P and none of set Q are present. Similarly, the blue-shaded region depicts the set Q – P, where all the elements of set Q and none of set P are present.

Learn more about Venn Diagram

Difference of Three Sets

If P, Q, and R are three non-empty sets, then the difference between the three of them can be depicted as P – Q – R.

This implies all the elements present in set P but not in sets Q and R. This is depicted in the Venn diagram below where the pink-shaded region depicts P – Q – R and the blue-shaded portion is the area not included in the difference.

Symmetric Difference of Sets

The symmetric difference of sets P and Q is expressed as P Δ Q and defined as 

P Δ Q = (P - Q) U (Q - P) 
OR
P Δ Q = (P ∪ Q) - (P ∩ Q)

Venn Diagram of Symmetric Difference of Sets

In the Venn diagram below, the pink-shaded portion represents P Δ Q.

Example: If P = {4, 5, 6, 7, 8} and Q = {6, 7, 8, 10}, find P Δ Q.

Solution:

• Step - 1: Find P - Q.
  P - Q = {4, 5, 6, 7, 8} - {6, 7, 8, 10} = {4, 5}
• Step - 2: Find Q - P.
  Q - P = {6, 7, 8, 10} - {4, 5, 6, 7, 8} = {10}
• Step - 3: Find P Δ Q = (P - Q) U (Q - P).
  P Δ Q = (P - Q) U (Q - P) = {4, 5} U {10} = {4, 5, 10}

Properties of Difference of Sets

If P and Q are two sets, then their difference has the following properties:

• P – Q = P ∩ Q'
• P – P = ∅
• P – ∅ = P
• ∅ – P = ∅
• P – Q = P, given that P ∩ Q = ∅
• P – Q = Q – P = ∅, if P = Q
• If P ⊂ Q, then P – Q = ∅
• n(P Δ Q) = n(P – Q) + n(Q – P)
• n(P Δ Q) = n(P U Q) – n(P ∩ Q)

Read More

• Types of Sets
• Operations on Sets
• Union of Sets

Solved Problems on Difference of Sets

Problem 1. Find the difference W – N, where W is the set of whole numbers and N is the set of natural numbers.

Solution:

Step 1. Write the given sets in set- builder form.

W = {0, 1, 2, 3, 4, 5,....,∞}
N = {1, 2, 3, 4, 5,....,∞}
W – N implies all elements of set W and none of set N.

Step 2. Write the difference in mathematical form: W – N = {0, 1, 2, 3, 4, 5,....,∞} – {1, 2, 3, 4, 5,....,∞}

Step 3. Strike out all the common elements in both W and N. {0, 1, 2, 3, 4, 5,....,∞} – {1, 2, 3, 4, 5,....,∞}.

All elements left in W represent the difference W – N.

Hence, W – N = {0}.

Problem 2. Prove  P – (Q ∪ R) = (P – Q) ∩ (P – R), if P = {1, 2, 4, 5}; Q = {2, 3, 5, 6} and R = {4, 5, 6, 7}.

Solution:

Let us consider the LHS first.

(Q ∪ R) = {x: x ∈ Q or x ∈ R}
⇒ Q ∪ R = {2, 3, 4, 5, 6, 7}.

Since P – (Q ∪ R) can be expressed as {x ∈ P: x ∉ (Q ∪ R)}.
⇒ P – (Q ∪ R) = {1}

Let us consider the RHS now.
P – Q is defined as {x ∈ P: x ∉ Q}
P = {1, 2, 4, 5}
Q = {2, 3, 5, 6}
⇒ P – Q = {1, 4}

Now, P – R is defined as {x ∈ P: x ∉ R}
⇒ P – R = {1, 2}

(P – Q) ∩ (P – R) = {x: x ∈ (P – Q) and x ∈ (P – R)}.
= {1}
∴ LHS = RHS

Hence verified.

Problem 3. If S and T are two sets, prove that: (S ∪ T) – T = S – T.

Solution:

Let us consider LHS first.

(S ∪ T) – T
= (S – T) ∪ (T – T)
= (S – T) ∪ ϕ (since, T – T = ϕ)
= S – T (since, x ∪ ϕ = x for any set)
= RHS

Hence proved.

Problem 4. If n(S) = 69, n(T) = 55, and n(S ∩ T) = 10, then what is n(S Δ T)?

Solution:

Since, n(S U T) = n(S) + n(T) - n(S∩ T)
= 69 + 55 - 10
= 114

According to the symmetric difference of sets,
n(S Δ T) = n(S U T) - n(S ∩ T)
= 114 - 10
n(S Δ T) = 104

Problem 5. If P, Q, R are three sets, such that P ⊂ Q, then prove that R – Q ⊂ R – P.

Solution:

Given, P ⊂ Q

To prove: R – Q ⊂ R – P

Let us consider, x ∈ R – Q
⇒ x ∈ R and x ∉ Q
⇒ x ∈ R and x ∉ P
⇒ x ∈ R – P

Thus, x ∈ R – Q ⇒ x ∈ R – P

This is true for all x ∈ R – Q
∴ R – Q ⊂ R – P

Hence proved.

Practice Questions on Difference of Sets

Problem 1: Find the difference X – Y, where X is the set of all integers from 1 to 10, and Y is the set of all odd numbers from 1 to 10.

Problem 2: Prove that A – (B ∪ C) = (A – B) ∩ (A – C), if A = {1, 2, 3, 4, 5}, B = {2, 4, 5}, and C = {3, 4, 6}.

Problem 3: If U = {1, 2, 3, 4, 5, 6}, A = {1, 2, 3}, and B = {3, 4, 5}, find the set (A ∪ B) – A.

Problem 4: If P = {2, 4, 6, 8}, Q = {1, 2, 3, 4, 5}, and R = {2, 3, 4, 5}, find P – (Q ∩ R).

Problem 5: If M = {a, b, c, d}, N = {b, c, e, f}, and O = {c, d, g, h}, find (M – N) ∪ (M – O).