Union of sets, Properties of union

Information processing in day-to-day life is expressed in set theory by the use of union sets whereby we can merge different sets together into a single set. For instance, consider two groups of friends: one team, basketball and another team soccer. In the same way, by uniting the two sets, we obtain a set that upon doing so collects all those that like both basketball and soccer. These properties contribute to the melding of sets being uniform and reliable, thus making the union of sets, a strong and valuable tool used with foresight to categorize and compare different compilations of objects in mathematical representation and others.

In this article, we will cover the concept of the union of sets and their properties. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. Over the last ten years of the JEE Main exam (from 2013 to 2023), a total of two questions have been asked on this concept, including one in 2014, and one in 2020.

Define Set

Sets are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U, V…...

What is a Union of Sets?

Let A and B be any two sets. The union of A and B is the set which consists of all the elements of A and all the elements of B, the common elements being taken only once. The symbol ‘∪’ is used to denote the union.

Symbolically, we write A U B = {x: x ∈ A or x ∈ B}.

If A = {1,3.5.7} and B = {2,4,6,8} then A∪B is read as A union B and its value is,

A∪B = {1,2,3,4,5,6,7,8}

Thus, from the above example, it is clear that a set that contains all the elements of set A and set B is called the union of set A and B.

Symbol of Union of Sets

The Union of the sets is represented using the symbol “∪”. It is placed between two sets whose union is to be found. We read this symbol as “union”. Example A∪B is read as A union B, furthermore we can also find the union of two or more sets as the union of set A, set B, and set C is represented as, A∪B∪C and is read as A union B union C.

Venn diagram of Union of Sets

The universal set (U) is usually represented by a rectangle and its subsets are usually represented by circles (or any other closed curve).

In the above-given diagram, the blue-coloured region shows the union of sets A and B. This further represents that the union between these sets includes all the elements that are present in A or B or both sets. Although the union operation between two sets has been used here, the Venn diagram is often used to represent the union between multiple sets, provided that the sets are finite.

For example, the set of natural numbers (N) is a subset of the set of whole numbers (W) which is a subset of integers (here integer(Z) is the universal set).
Properties of union

1. Commutative Property: This signifies that the union of sets is independent of interchangeable properties.

A ∪ B = B ∪ A

2. Associative Property: This signifies that the union of three sets can be interchangeable.

( A B ) C = A ( B C)

3. Law of identity element: When we take the union of a finite set from a null set, the original set comes (φ is the identity of Null Set).

A φ = A

4. Idempotent Property: This signifies that the union of the same set is itself.

A A = A

5. Property of universal set: When we take union from a universal set then a universal set will come.

U A = U

Note: If A is a subset of B, then A ∪ B = B

Solved Examples Based On Union of sets and their properties:

Example 1: If set A={1,3,5} and B={3,5,7}. Also and . Then which of the following is true?

1)

2)

3)

4)

Solution:

In this Question,

As A U B = B U A

So, P = Q.

Hence, the answer is the option 1.

Example 2: If A={5, 6, 7} and B={ }, then the value of A B is

1) { }

2)

3) {5,6,7}

4) {5}

Solution:

A U φ = A

Using this property, the union in this question will be {5,6,7}.

Hence, the answer is the option 3.

Example 3:

1)

2)

3)

4)

Solution:

As we have learnt

(A ∪ B) ∪ C = A ∪ (B ∪ C)

In this question,

A U B = P

So, P U C = (A U B) U C ....(i)

B U C = Q

So, A U Q = A U (B U C) ....(ii)

From (i) and (ii) and Associative property

P U C = A U Q.

Hence, the answer is the option 1.

Example 4: If n(U) = 20, n(A)=12, n(B)=9, where U is the universal set, then

Solution:

n(A U B) = n(A) + n(B) - n(A B) = 12 + 9 - 4 = 17

n(A U B)' = n(U) - n(A U B) = 20 - 17 = 3

Hence, the answer is 3.

Example 5: Find the union of the sets {3,7,9}, {5,8,1} and {2,8,3}.

1) {1,2,3,4,5,8,7,9}

2) {1,2,3,5,7,8,9}

3) {1,2,3,4,7,8,9}

4) {1,2,5,7,8,9}

Solution:

The union of these three sets is the set containing all the elements in these three sets. Option (2) is correct

Options (1) and (3) have 4, which is not present in any of the given sets.

Option (4) is missing 3.

Hence, the answer is the option 2.

Summary

From a mathematician’s point of view, the union of sets is a common and very useful operation widely used in solving many problems. Through the understanding of the union operation, it has been noted that through the properties of commutativity, associativity, and identity property. In the social context, the union operation enables one to bring together multiple social groups to become part of a larger group, while in data, it helps in assembling entire sets or data containing all the pertinent aspects. With this kind of characteristic in mind, the role and usefulness of the union of sets extend far and wide and the following discussions delve into further explanation of this concept.