Definition of Union of Sets:
Union of two given sets is the smallest set which contains all the elements of both the sets.
To find the union of two given sets A and B is a set which consists of all the elements of A and all the elements of B such that no element is repeated.
The symbol for denoting union of sets is ‘∪’.
For example;
Let set A = {2, 4, 5, 6}
and set B = {4, 6, 7, 8}
Taking every element of both the sets A and B, without repeating any element, we get a new set = {2, 4, 5, 6, 7, 8}
This new set contains all the elements of set A and all the elements of set B with no repetition of elements and is named as union of set A and B.
The symbol used for the union of two
sets is ‘∪’.
Therefore,
symbolically, we write
union of the two sets A and B is A ∪ B which means A union B.
Therefore, A ∪ B = {x : x ∈ A or x ∈ B}
Solved examples to find union of two given sets:
1. If A = {1, 3, 7, 5} and B = {3, 7, 8, 9}. Find union of two set A and B.
Solution:
A ∪ B = {1, 3, 5, 7, 8, 9}
No element is repeated in the union of two sets.
The common elements 3, 7 are taken only once.
2. Let X = {a, e, i, o, u} and Y = {ф}. Find union of two given sets X and Y.
Solution:
X ∪ Y = {a, e,
i, o, u}
Therefore,
union of any set with an empty set is the set itself.
3. If set P = {2, 3, 4, 5, 6, 7}, set Q = {0, 3, 6, 9, 12} and set R = {2, 4, 6, 8}.
(i) Find the union of sets P and Q
(ii) Find the union of two set P and R
(iii) Find the union of the given sets Q and R
Solution:
(i) Union of sets P and Q is P ∪ Q
The smallest set which contains all the elements of set P and all the elements of set Q is {0, 2, 3, 4, 5, 6, 7, 9, 12}.
(ii) Union of two set P and R is P ∪ R
The smallest set which contains all the elements of set P and all the elements of set R is {2, 3, 4, 5, 6, 7, 8}.
(iii) Union of the given sets Q and R is Q ∪ R
The smallest set which contains all the elements of set Q and all the elements of set R is {0, 2, 3, 4, 6, 8, 9, 12}.
Notes:
A and B are the
subsets of A ∪ B
The union of sets is commutative, i.e., A ∪ B = B ∪ A.
The operations are performed when the sets are
expressed in roster form.
Some properties of the operation of union:
(i) A∪B = B∪A (Commutative law)
(ii) A∪(B∪C) = (A∪B)∪C
(Associative law)
(iii) A ∪ ϕ = A (Law of identity element, is the
identity of ∪)
(iv) A∪A = A
(Idempotent law)
(v) U∪A = U
(Law of ∪) ∪ is the universal set.
Notes:
A ∪ ϕ = ϕ ∪ A = A i.e. union of any set with the empty set is always the set itself.
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