A set is a group or collection of different items, objects, or numbers. These items are called the elements of the set. A set can include anything—such as numbers, letters, colors, days of the week, or even types of vehicles. For example, the group of numbers 1, 2, 3, 4, and 5 can be written as a set:
A = {1, 2, 3, 4, 5}
Sets are usually written inside curly brackets { } and are named using capital letters like A, B, or C.

There are two common ways to write sets:

1. Roster form – listing all the elements, like: A = {red, blue, green}
2. Set-builder form – describing the rule of the set, like: A = {x | x is a prime number less than 10}

In this mathematics article, we will learn the definition of union and intersection of sets. We will also solve problems based on the union and intersection of sets.

What are Sets?

In maths, a set is a group or collection of different objects, called elements. These elements can be anything—numbers, letters, shapes, or even other sets. All the items in a set are unique, which means no element is repeated.

We write a set by placing its elements inside curly brackets { }. For example, the set {1, 2, 3} means it contains the numbers 1, 2, and 3.

Sets can be of two types:

• A finite set has a fixed number of elements. Example: {a, b, c}.
• An infinite set has no end. Example: the set of all natural numbers {1, 2, 3, 4, ...} goes on forever.

Sets are useful in many areas of mathematics, such as comparing groups, understanding data, and solving problems using Venn diagrams and operations like union and intersection.

The concept of sets allows us to define relationships between objects and perform operations on sets. These operations include union, intersection, difference, and complement. We will learn about union and intersection of sets in detail.

Types of Sets

These are some of the common types of sets in mathematics. Each type has its own characteristics and applications.

• Empty Set or Null Set: This type of set has no elements and is denoted by the symbol ∅ or {}. It is used to represent the absence of elements in a set.

• Singleton Set: A singleton set contains only one element. For example, {1} is a singleton set with the element 1.

• Equal Sets: Two sets are considered equal if they have the exact same elements, regardless of the order or repetition of elements.

• Subset: A set A is a subset of another set B if every element of A is also an element of B. The symbol ⊆ is used to denote subset.

• Proper Subset: A proper subset is a subset that is not equal to the original set. In other words, all the elements of the proper subset are also elements of the original set, but the original set has additional elements.

• Universal Set: The universal set, denoted by the symbol U, is the set that contains all the elements under consideration in a particular context.

• Disjoint Sets: Disjoint sets are sets that have no common elements. In other words, the intersection of disjoint sets is an empty set.

• Power Set: The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself.

These are some of the common types of sets in mathematics. Each type has its own characteristics and applications in various mathematical contexts.

What is Union and Intersection of Sets?

Union and intersection of sets is the set operations that are used in set theory. Operations on sets are mathematical procedures that allow us to combine, compare, and manipulate sets. These operations help us analyze the relationships between sets and extract useful information.

The union of any two or more sets results in a completely new set that contains a combination of elements that are present in both those two or more given sets. The intersection of any two or more sets is the set that contains all the elements that are common to both those two or more given sets.

Here are some other common operations on sets:

• Complement: The complement of a set consists of all the elements not present in the set, considering a universal set as a reference.

• Cartesian Product: The Cartesian product of two sets A and B is the set of all possible ordered pairs where the first element comes from A and the second element comes from B.

• Power Set: The power set of a set is the set of all possible subsets of that set, including the empty set and the set itself.

Union of Set

Union of sets is the set of elements which are a combination of elements that are present in both the given sets. The symbol \(\cup\) is used to represent the union of two sets. For example, the union of two sets \(A\) and \(B\) is written as \(A\cup B\).

• For two disjoint sets \(A\) and \(B\), their union is \(A\cup B\) is the combination of all the elements in both the sets.

• Set \(A\) is a subset of a set \(B\) if all elements of \(A\) are also elements of \(B\). For two sets \(A\) and \(B\), we have \(A\subset(A\cup B)\), and \(B\subset(A\cup B)\).

Example of Union of Sets

Let's consider a scenario where we have two sets of students involved in different extracurricular activities in a school. Set A represents students in the Chess Club, and Set B represents students in the Drama Club.

A = {John, Emma, Sarah, Michael}
B = {Emma, Michael, Olivia, Ethan}

To find the union of these sets, we combine all the unique students from both sets.

A ∪ B = {John, Emma, Sarah, Michael, Olivia, Ethan}

In this case, the union of sets A and B gives us a new set that contains all the students who participate in either the Chess Club or the Drama Club, without duplicating any names.

Properties of the Union of Sets

The properties of union of sets are tabulated below:

Applications of Union of Sets:

1. Combining Data from Two Groups:
   If you have a list of students who play football and another list of students who play cricket, the union gives all students who play either football, cricket, or both.

2. Survey Results:
   In surveys, if some people like tea and others like coffee, the union shows all people who like either tea or coffee or both.

3. Library Books:
   If one shelf has fiction books and another has science books, the union tells you all the different books available across both shelves.

4. Database Merging:
   In computer science, if two different databases store names, the union combines them without repeating any name.

Intersection of Sets

Intersection of sets is the set of elements which are common to both the given sets. The symbol \(\cap\) is used to represent the intersection of two sets. For example, the intersection of two sets \(A\) and \(B\) is written as by \(A\cap B\).

• For two disjoint sets \(A\) and \(B\), their intersection is \(A\cap B = \phi\).
• For two sets \(A\) and \(B\), we have \((A\cap B)\subset A\), and \((A\cap B)\subset B\).

Example of Intersection of Sets

Let's again take the sets A and B. They represent the Chess Club and Drama Club students, respectively.

A = {John, Emma, Sarah, Michael}
B = {Emma, Michael, Olivia, Ethan}

To find the intersection of these sets, we identify the common students who are members of both clubs.

A ∩ B = {Emma, Michael}

The intersection of sets A and B is the set of students who are members of both the Chess Club and the Drama Club. Emma and Michael are the only students present in both sets.

Properties of the Intersection of Sets

The properties of intersection of sets are tabulated below:

Applications of Intersection of Sets:

1. Finding Common Members:
   If one group of students takes mathematics and another takes science, the intersection will show the students who take both subjects.

2. Venn Diagram Analysis:
   In logic and data science, the intersection helps identify overlapping parts of different categories.

3. Inventory Management:
   A store may use the intersection to find products that are in demand in both summer and winter, helping with stock planning.

4. Social Media:
   Platforms can use intersection to find users who like two different pages or topics, for better content targeting.

Union and Intersection of Sets Venn Diagram

Venn Diagrams refer to the diagrams that are used to represent the relationship between the given set operations. Any set operation can be represented by using a Venn diagram. Venn diagrams represent each set using circles.

Let’s see how to use the Venn diagram to represent the union of two sets. For this, we first need a universal set, of which the two given sets \(A\) and \(B\) are the subsets. The following Venn diagram represents the union between the sets \(A\) and \(B\).

In the above Venn diagram, the shaded region shows the union of sets \(A\) and \(B\).

In the same way, we can draw a Venn diagram for the union of three sets as shown below:

In the above Venn diagram, the shaded region shows the union of three sets \(A\), \(B\) and \(C\).

Now, let’s see how to use the Venn diagram to represent the intersection of two sets. For this, we first need a universal set, of which the two given sets \(A\) and \(B\) are the subsets. The following Venn diagram represents the intersection between the sets \(A\) and \(B\).

In the above Venn diagram, the shaded region shows the intersection of sets \(A\) and \(B\).

In the same way, we can draw a Venn diagram for the intersection of three sets as shown below:

In the above Venn diagram, the orange colored region shows the intersection of three sets \(A\), \(B\) and \(C\).

Union and Intersection of Complement Sets

The complement of a set is the set of elements that are not present in the original set but belong to the universal set.

Union of Complement Sets

The union of complement sets involves combines the elements that are not in either of the original sets.

A' ∪ B' = {x | x is in the universal set but not in either A or B}

Properties of Union of Complement Sets

De Morgan's Law: The union of the complement sets is equal to the complement of the intersection of the original sets. (A ∩ B)' = A' ∪ B'.

Intersection of Complement Sets

The intersection of complement sets includes the elements that are absent in both of the original sets.

A' ∩ B' = {x | x is in the universal set and not in either A or B}

Properties of Intersection of Complement Sets:

De Morgan's Law: The intersection of the complement sets is equal to the complement of the union of the original sets. (A ∪ B)' = A' ∩ B'.

Union and Intersection of Finite Sets 

We can perform union and intersection operations on both finite and infinite sets.

Union of Finite Sets: The union of two finite sets combines all the unique elements from both sets. This results in a set that contains all the elements from both sets without any duplicates.

For example, let's consider two finite sets: A = {1, 2, 3} B = {3, 4, 5}

The union of sets A and B is: A ∪ B = {1, 2, 3, 4, 5}

Thus, the union operation combines the elements of both sets. It results in a new set with all the distinct elements.

Intersection of Finite Sets: The intersection of two finite sets identifies the common elements shared by both sets.

Continuing with the previous example, the intersection of sets A and B is: A ∩ B = {3}

The intersection operation reveals that the common element between sets A and B is 3, as it is the only element present in both sets.

Union and Intersection of Infinite Sets

The union and intersection operations can also be performed on infinite sets, with some interesting outcomes.

For example, let's consider two infinite sets: A = {1, 2, 3, ...} (the set of all natural numbers) B = {2, 4, 6, ...} (the set of all even numbers)

The union of sets A and B is: A ∪ B = {1, 2, 3, 4, 5, 6, ...} (the set of all natural numbers and even numbers)

In this case, the union operation combines all the elements from both sets, resulting in a set that contains all the natural numbers and even numbers.

The intersection of sets A and B is: A ∩ B = {2, 4, 6, ...} (the set of all even numbers)

The intersection operation reveals that the common elements between sets A and B are all the even numbers, as they are present in both sets.

Difference Between Union and Intersection of Sets

The difference between union and intersection of sets are tabulated below:

Union and Intersection of Sets Summary

• The union of two sets \(A\) and \(B\) is defined as the set of elements that are included either in the set \(A\) or set \(B\), or both \(A\) and \(B\). The symbol \(\cup\) is used to represent the union of two sets.
• The intersection of two sets \(A\) and \(B\) is defined as the set of elements that belongs to both the sets \(A\) and \(B\). The symbol \(\cap\) is used to represent the intersection of two sets.
• The formula for union of \(A\) and \(B\) two sets is \(A\cup B\) = {\(x: x\in A\) or \(x\in B\)}.
• The formula for intersection of \(A\) and \(B\) two sets is \(A\cap B\) = {\(x: x\in A\) and \(x\in B\)}.
• The union of sets rejects the identical values from the set.
• The intersection of sets is an associative operation which includes the common values from the set.

Union and Intersection of Sets Solved Example

Example 1: Find the union and intersection of the sets \(A={2,3,5,6,7}\), and \(B={4,5,7,8}\).

Solution: Given \(A={2,3,5,6,7}\) and \(B={4,5,7,8}\).

\(A\cup B = {2,3,4,5,6,7,8}\).

\(A\cap B = {5,7}\).

Example 2: Find the union and intersection of the sets given below, \(P={a,e,i,o,u}\), \(Q={p,q,r,s,t}\), and \(R={j,k,l,m,n}\).

Solution: Given \(P={a,e,i,o,u}\), \(Q={p,q,r,s,t}\), and \(R={j,k,l,m,n}\).

Union of the given sets is:

\(P\cup Q\cup R={a,e,i,o,u,p,q,r,s,t,j,k,l,m,n}\).

Intersection of the sets is:

\(P\cap Q\cap R=\phi\).

The intersection of the sets \(P\), \(Q\), and \(R\) is a null set (empty set) since no element is common among the three sets.

Example 3: Prove that the associative property for the union of sets, the sets \(A={a,b,c,d,e}\), \(B={d,e,f,g,h}\), and \(C={g,h,i,j,k}\).

Solution: Given \(A={a,b,c,d,e}\), \(B={d,e,f,g,h}\), and \(C={g,h,i,j,k}\).

Associative property for the union of three sets is \((A\cup B)\cup C=A\cup (B\cup C)\).

For LHS, \(A\cup B={a,b,c,d,e,f,g,h}\)

\((A\cup B)\cup C={a,b,c,d,e,f,g,h,i,j,k}\) …….(equation 1)

For RHS, \(B\cup C={d,e,f,g,h,i,j,k}\)

\(A\cup (B\cup C)={a,b,c,d,e,f,g,h,i,j,k}\) …….(equation 2)

From equation 1 and 2 , we have

LHS = RHS

Hence, \((A\cup B)\cup C=A\cup (B\cup C)\).

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