In Set Theory we usually perform different operations on sets such as intersection, union, complement. The Difference of Sets is also a similar kind of operation which we perform on sets. You will understand the difference between intersection and difference of sets clearly after going through this article. Check out **Set Theory** to be clear with the concepts of Sets Operations.

## How to find the Difference of Sets?

In general, the Difference between the Two Sets is the Set of elements present in A but not in B. It is represented as A – B. You can see the difference in the orange shaded region of the below Venn diagram. In the same way, the region shaded in violet indicates the difference between B and A.

### Identities Involving Difference of Sets

- If Set A and B are equal then A-B = A-A = ϕ
- If you subtract an empty set from a Set then the result is the Set itself i.e. A – ϕ = A.
- In the Similar Way, if you subtract a Set from an Empty Set then the result is an Empty Set i.e. ϕ – A = ϕ
- If you subtract a Superset from Subset the result is an empty set i.e. A – B = ϕ if A ⊂ B
- If Two Sets A and B are disjoint then A – B = A, B – A = B

### Solved Examples for finding Difference of Sets

1. If A = {4, 5, 6} and B = {7, 8, 9}. Find the Difference between Sets A and B and B and A?

Solution:

Given A = { 4, 5, 6}

B = {7, 8, 9}

A-B = {4, 5, 6} – { 7, 8, 9}

= { 4, 5, 6}

B-A = {7, 8, 9} – {4, 5, 6}

= {7, 8, 9}

Since two sets A, B are disjoint the difference between A and B yields A and difference between B and A gives B.

2. Let A = {c, d, e, f, g, h, i} and B = {b, d, f, g, i, h} find A-B and B-A?

Solution:

Given A = {c, d, e, f, g, h, i}

B = {b, d, f, g, i, h}

A-B = {c, d, e, f, g, h, i} – {b, d, f, g, i, h}

= { c, e}

The elements c, e belong to Set A but not B

B-A = {b, d, f, g, i, h} – {c, d, e, f, g, h, i}

= {b}

Thus, the element b belongs to Set B but not A.

3. Given Sets A = {3 , 4 , 8 , 9 , 11 , 12 } B = {3 , 4 , 8 , 9 , 11 , 12 }. Find the difference between them i.e. A-B?

Solution:

A = {3 , 4 , 8 , 9 , 11 , 12 }

B = {3 , 4 , 8 , 9 , 11 , 12 }

A-B = A- A( since both the sets are equal)

= ϕ or {}

When you subtract two equal sets the difference between them will be an Empty Set.

4. Given three sets P, Q and R such that:

P = {x : x is a natural number between 12 and 18},

Q = {y : y is a even number between 14 and 20} and

R = {5, 9, 10, 12, 18, 20}

(i) Find the difference between two sets P and Q

(ii) Find Q – R

(iii) Find R – P

(iv) Find Q – P

Solution:

Given P = {x : x is a natural number between 12 and 18}

P = { 13, 14, 15, 16, 17}

Q = {y : y is a even number between 14 and 20}

Q = { 16, 18}

R = {5, 9, 10, 12, 18, 20}

(i) Difference between two sets P and Q i.e. P-Q

P-Q = { 13, 14, 15, 16, 17} – { 16, 18}

= {13, 14, 15, 17}

Elements 13, 14, 15, 17 are there in P but not in Q.

(ii)Q – R

Q = { 16, 18}

R = {5, 9, 10, 12, 18, 20}

Q – R = { 16, 18} – {5, 9, 10, 12, 18, 20}

= {16}

16 is the element that is present in Q but not in R.

(iii) R – P

R = {5, 9, 10, 12, 18, 20}

P = {13, 14, 15, 16, 17}

R – P = {5, 9, 10, 12, 18, 20} – {13, 14, 15, 16, 17}

= { 5, 9, 10, 12, 18, 20}

(iv) Q – P

Q = { 16, 18}

P = {13, 14, 15, 16, 17}

Q – P = { 16, 18} – {13, 14, 15, 16, 17}

= { 18}

18 is the element that is present in Q but not in P.