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.