To solve problems on the union of sets you need to be aware of what is meant by a union. Get the Formulas associated with the Union of Sets. For better understanding, we have listed solved examples on finding the union of sets. After going through this article, you can get a fair idea of how to find the union of two or more sets. To know more on Operations of Sets check out the Set Theory and learn the concepts effectively.
Union of Sets
Union of Two Sets A and B is the Set of elements in Set A or B or in both. It is represented as A U B and is areas as A union B or Union of A and B.
(i) Union of Disjoint Sets
If A and B are two finite sets and if A ∩ B = ∅ then
n(A U B) = n(A) +n(B)
Union of Disjoint Sets represented using the Venn Diagram as such. A ∩ B = ∅ since no two sets have common elements.
(ii) Union of Two Sets
If A and B are two finite sets then A union B is given by
n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
You can simply say that the Union of A and B is nothing but the summation of cardinal numbers of Set A and B minus Intersection of them.
In the above diagram, the three disjoint sets are A – B and B – A, A ∩ B their sum represents the A U B. Thus, we can say
n (A ∪ B) = n (A – B) + n(B – A) + n(A ∩ B)
(iii) Union of Three Sets
If A, B, C are three finite sets then n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)
From the Venn diagram, it is quite clear that the Union of Three Sets is the summation of cardinal numbers of Set A, Set B, Set C, and the common elements of three sets excluding the common elements of sets taken in pairs.
Solved Examples on Union of Sets
1. A = {0, 1, 3, 5, 7, 9}, B = {2, 4, 6, 8} and C = {1, 5, 7, 9}. Find (A ∪ B) ∪ C?
Solution:
(A U B) = {0, 1, 3, 5, 7, 9} U {2, 4, 6, 8}
= { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
(A ∪ B) ∪ C = { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9} U {1, 5, 7, 9}
= { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
2. X = {3, 4, 5, 6}, Y = {2, 5, 7} and Z = {4, 5, 6}. Verify X U Y = Y U X
Solution:
X U Y = {3, 4, 5, 6} U {2, 5, 7}
= { 2, 3, 4, 5, 6, 7}
Y U X = { 2, 5, 7} U {3, 4, 5, 6}
= { 2, 3, 4, 5, 6, 7}
Therefore X U Y = Y U X.
3. Let A = {x : x is a natural number and a factor of 12} and B = {x : x is a natural number and less than 4}. Find A ∪ B?
Solution:
A = { 1, 2, 3, 4, 6, 12}
B = { 1, 2, 4}
A U B = { 1, 2, 3, 4, 6, 12} U { 1, 2, 4}
= { 1, 2, 3, 4, 6, 12}