Examples of Intersection of Sets With Answers

The method of extracting the common elements of two or more sets into a single set is called the intersection of sets. It is represented by the symbol ‘∩’, and the word “and” is used in its mathematical definition. Common elements are written only once.

In this article, we will look at examples of the intersection of sets with answers involving two and three sets, and detailed solutions.

What Is the Intersection of Sets?

The intersection of two sets A and B forms a set consisting of the common elements of set A and set B, and the other elements are not included in this set.

It is represented as A ∩ B.

In mathematical notation,

A ∪ B = {x | x ∈ A and x ∈ B}

Here, the word “and” means that elements belong to both A and B.

For Example,

Let:

A={2,4,6}

and

B={0,4,6,8,10}

According to the definition of the intersection of sets, here is a set that has the common elements of both set A and set B.

A ∩ B ={4,6}

Do you take a look at 4 and 6?  These are written only once because they are in both sets.

How to Find the Intersection of Two Sets

Here are some simple steps that will help in finding the intersection of any two sets. After these steps, you will get solved examples of the intersection of different sets for better understanding.

• 1. Write the elements of the first set that are also in the second set.
• 2. Don’t write the elements of the second set that are already in the first set.
• 3. Arrange the elements in a suitable order, usually from smallest to largest.

Examples of Intersection of Sets with Answers

Example 1:

These sets have common elements, and their intersection will list the common elements only once.

Let

A={1,3,5,7}

B={1,2,6,8}

Then,

A∩B={1}

Example 2:

Here are the sets that have disjoint elements. There is no common element in both sets, so the intersection will be the empty set.

Let:

P={10,20,30}

Q={50,60,70}

Then,

P∩Q={ }

Example 3:

Here, set M is a subset of set N, and the intersection of these sets will contain all the elements of the subset, i.e set M.

Let:

M={a,b,c}

N={a,b,c,d,e,f}

Then,

M∩N= {a,b,c}

You will note that if M⊆N, then M∩N=M.

Example 4:

Both sets have letters.

Let:

U={a,e,i,o,u}

V={a,b,d,e}

Then,

U∩V= {a,e}

Example 5:

Both sets have names.

Suppose:

A={Ali, Ahmad, Anis}

B={Ben, David, Elena}

Now,

A∩B={ }

Example 6:

 Intersection of three sets with disjoint elements.

Let:

A={1,2,3}

B={4,5,6}

C={7,8,9}

First, take the intersection of two sets

A∩B={ }

Now,

(A∩B)∩C={ }

Example 7:

Sets of Multiples

Let:

X={2,4,6,8}

Y={3,6,9,12}

Z={4,8,12,16}

Then:

X∩Y∩Z={ }

Example 8:

Intersection with an empty set

Let:

A={4,6,10,14}

Then:

A∩∅=∅

Example 9:

Intersection of Odd and even numbers

Let:

E={2,4,6,8}

and

O={1,3,5,7,9}

Then:

E∩O={ }

Example 10:

Let:

A={x∈N∣1≤x≤4}

and:

B={x∈N∣4≤x≤7}

Writing the sets in simple form:

A={1,2,3,4}

B={4,5,6,7}

Therefore:

A∩B={4}

Practice Questions

Q1. If A={1,2,3} and B={2,3,4}, then find A∩B.

Q2. Find: P∩Q where:P={a,c,e},Q={b,d,f}.

Q3. Find: X∩Y where: X={10,20,30,40},Y={20,40}.

Q4. If: A={1,3,6},B=∅ find A∩B.

Q5. Let: A={12,14,16},B={1,13,15},C={6,17,18} Find A∩B∩C.

Venn Diagrams

Here are some examples of the intersection of sets with Venn diagrams for the students’ ease.

Properties of Intersection

Here are some common and important properties of the Intersection of sets that will help in solving questions based on Intersection.

1. Commutative Property

A∩B=B∩A

2. Associative Property

(A∩B)∩C=A∩(B∩C)

3. Identity Property

A∩∅=∅

4. Idempotent Property

A∩A=A

You can also visit Examples of Union of sets for more practice.

FAQs

Q1. What is the Intersection of two sets?

The Intersection of two sets contains every unique element of both sets and doesn’t repeat the same element. It is represented by:

A∩B

Q2. Can an element appear twice in an intersection?

No. An element appears only once in an intersection.

Q3. What is the intersection of a set and the empty set?

When we take the Intersection of any set, like set A, with the empty set, we get the empty set as a result.

A∩∅=∅

Q4. Is Intersection the same as addition?

No. Intersection is not the same as ordinary addition.

Q5. Which symbol represents union and which represents intersection?

The symbol “∪” represents union, while “∩” represents intersection.

Conclusion

In this article, we have solved examples of the Intersection of sets with answers for students’ practice. The Intersection of different types of sets is solved one by one. After this article, any student can easily solve the Intersection of sets.