Union of Sets

So, in mathematics, there are a whole bunch of set types: ordered sets, unordered sets, ordinals, cardinals, ordinal numbers, cardinal numbers, and even an uncountable set of all sets: the universe of sets.

So the question is, what's the point of having so many different set types, and why would anyone want to use them? 

Union of sets

Union of two sets is the least set which comprises all the elements of both the sets.

To find the union of two sets, we take X and Y, which contains all the elements of X and all the elements of Y such that no element is repeated.

The symbol for representing the union of sets is ‘∪’.

For Example:

Let us assume that set A = {2, 4, 5, 6}

and set B = {4, 6, 7, 8}

Now, we take each and every element of both the sets X and Y, making sure that no element is repeated. We are now left with a new set = {2, 4, 5, 6, 7, 8}

This new set comprises all the elements of set X and all the elements of set Y with no recurrence of any element and is named as the union of set X and Y.

The symbol used for this union of two sets is ‘∪’.

Therefore, we accomplish from the definition of the union of sets that A ⊆ A U B, B ⊆ A U B.

Henceforth, representatively, we write the union of the two sets X and Y as X ∪ Y, which means X union Y. 

Therefore, X ∪ Y = {x : x ∈ X or x ∈ Y} 

Union of Sets Examples

1. If X = {1, 3, 7, 5} and Y = {3, 7, 8, 9}. What is the union set of X and Y?

Solution: X ∪ Y = {1, 3, 5, 7, 8, 9} 

It is to note that there is no repetition of any element in the union of two sets here. The elements which are common, like 3 7, are only taken once. 

2. Let A = {a, e, i, o, u} and B = {ф}. Find the union of two given sets, A and B. 

Solution: A ∪ B = {a, e, i, o, u} 

Therefore, if any set is empty, its union of sets is the set itself. 

3. If set B = {2, 3, 4, 5, 6, 7}, set C = {0, 3, 6, 9, 12} and set D = {2, 4, 6, 8}.

(i) Find the union of sets B and C.

(ii) Find the union of two sets, B and D.

(iii) Find the union of the given sets C and D.

Solution: Union of sets B and C is B ∪ C.

(i) The smallest set, which comprises all the elements of set B and every element of set C is {0, 2, 3, 4, 5, 6, 7, 9, 12}.

(ii) Union of two sets B and D is B ∪ D.

The smallest set, which comprises every element of set B and every element of set D, is {2, 3, 4, 5, 6, 7, 8}.

(iii) Union of the given sets C and D is C ∪ D.

The smallest set, which comprises every element of set C and every element of set D is {0, 2, 3, 4, 6, 8, 9, 12}.

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 ∪) 

Union Venn Diagram

(image will be  uploaded soon)

From the above A union B Venn diagram, the following theorems are evident:

(i) A ∪ A = A                    (Idempotent theorem)

(ii) A ⋃ U = U                      (Theorem of ⋃) U here is known to be the universal set.

(iii) If A ⊆ B, then A ⋃ B = B

(iv) A ∪ B = B ∪ A              (Commutative theorem)

(v) A ∪ ϕ = A                       (Theorem of identity element depicts the identity of ∪) 

(vi) A ⋃ A' = U                    (Theorem of ⋃) U here is known to be the universal set.