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 of 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 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

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}.

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

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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.


Set Operations

The union of two sets is a set covering every element that is present in A or B or possibly both. Let’s take an instance of set {1,2} ∪ {2,3} =  {1,2,3}{1,2}∪{2,3}={1,2,3}.

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The shaded area represents the set B∪A. 

Similarly, the union of three or more sets can also be defined. In particular, if A1, A2, n sets, their union A1∪A2∪A3 is a set that comprises all the elements that are at the minimum in one of the sets. 

For instance, if A1={a,b,c},A2={c,h},A3={a,d} then ⋃iAi=A1∪A2∪A3={a,b,c,d}. We can in the same way define the union of infinite number of sets. 

The intersection of two sets A and B, represented by A∩B which consists of all elements that are both in A and B.
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The shaded area represents the set B∩AB∩A.

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The shaded area depicts the set A∩B∩CA∩B∩C.
The accompaniment of a set A is denoted by Ac or A¯, is the set of all elements that are present in the universal set S but are absent in A. 

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The set A−B comprises elements that are in A but are absent in B. For example if A={1,2,3} and B={3,5} }, then A−B={1,2}. 

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Let’s assume two sets A and B which will disjoint if they lack any shared elements. i.e. A∩B=∅A∩B=∅.

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Consider the earth's surface as our sample space, we would have to partition it into different continents. Likewise, when it comes to a country, it would be partitioned to different provinces. In general, a group of nonempty sets A1, A2 is a partition of a set A if they are disjoint and their union is called A.

FAQ (Frequently Asked Questions)

1. How Many Types of Sets are There? 

There are many types of sets in the set theory:

1. Singleton set

2. Finite Set

3. Infinite set

4. Equal set

5. Null set/ empty set

6. Subset

7. Proper set

8. Improper set

9. Power set

10. Universal set

2. Can an Element in a Set Be Negative in Value?

A set can have any number of non-negative quantities of elements, extending from none (the empty set or null set) to infinitely as many as needed. The number of elements in a set is known as the cardinality, and they can extend from zero to denumerably infinite (for the sets of natural numbers, integers, or rational numbers) to non-denumerably infinite for the sets of irrational numbers, real numbers, imaginary numbers, or complex numbers).