 # 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 ∪)

### Image will be added 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.

### 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}.

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.

The shaded area represents the set B∩AB∩A.

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.

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}.

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

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.

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).