Posted by Anjali Kaur on Jul 07, 2020
Sets and Venn diagram

Sets and Venn diagram

We have seen the introduction to sets theory, types of sets, and different types of numbers. Before understanding the usage of sets and Venn diagram in sets theory, we will learn the types of intervals used in sets theory. Let’s learn.

What is an interval?

An interval is a set that consists of all real numbers between a given pair of numbers.

There are two types of intervals used under sets theory:

  1. Open Interval: Remember open intervals are those where elements are not included or they are excluded, there is no ‘=’ sign used and it is represented by ‘()’ parenthesis. Examples of open interval:
    • {x: a<x<b} this interval represents elements a and b in an open interval, as they are not included in this set; (a,b)
    • {x: a≤x<b} this set says element ‘a’ is included (there is less than equal to sign) and b is excluded; [a,b)
    • {x: a<x≤b} this set says element ‘b’ is included (Just remember wherever you find equal to sign in an interval, that will be included) and a is excluded; (a,b]
  2. Close Interval: Remember close intervals are those where both the elements are included (Like a closed-door, whatever is inside is included), there is always ‘=’ sign used on both the ends of the interval and it is represented with ‘[]’. Example of close interval:
    • {x: a≤x≤b} this interval represents elements a and b in a closed interval as they are included in this set; [a,b]

What is a Venn diagram?

Sets and Venn diagram are represented by circles drawn inside a rectangle representing the universal set. Like this;

The overlapping region of two circles represents the intersection of the two sets.

Two circles together represent the union of the two sets. As shown below:

Complement of a Set

If A is any set, then its complement is represented by A’

A’ = U – A, where U is a universal set that comprises of every element. The region outside the circle represents the complement of the set.

Union and Intersection of a Set

Union of a set means included everything once. For example;

A = {1,2,3,4}

B = {3,4,5,6}

A U B = {1,2,3,4,5,6}; so union includes every element of a sets once.

(A U B is also termed as elements of a set in A or in B)

The intersection of a set means include only common elements in a set. For example;

A = {1,2,3,4}

B = {3,4,5,6}

A ∩ B = {3,4}; so intersection includes only the common elements of sets.

(A ∩ B is also termed as elements of a set in A and B both)

Disjoint Sets

The sets that do not have any element in common. For example;

A = {1,3,5}

B = {2}

A ∩ B =  

Symmetric Difference

Symmetric difference of two sets is the set of elements which are in either of the sets and not in their intersection. The symmetric difference of the sets A and B is commonly denoted by ‘△’.

(A-B) U (B-A) = A △ B

Important formulas

  1. A U A’ = U
  2. A ∩ A’ =  
  3. A’ = U – A
  4. n(A U B) = n(A) + n(B) – n(A ∩ B)
  5. n(A U B U C) = n(A) + n(B) + n(C) – n(A∩ B) -n(A ∩ C) – n(B ∩ C) + n(A∩B∩C)
  6. n(A U B’) = n(A-B) = n(A) – n(A ∩ B)
  7. n(A’ ∩ B) = n(B-A) = n(B) – n(A ∩ B)

Test Yourself

Question. In a group of 800 people, 550 can speak Hindi, and 450 can speak English. How many can speak both Hindi and English?

Solution. Let ‘A’ as Hindi speaking people and ‘B’ as English speaking people.

n(A U B) = 800

n(A) = 550

n(B) = 450

We have to find n(A ∩ B)

Use the below formula,

n(A U B) = n(A) + n(B) – n(A ∩ B)

800 = 550 + 450 – n(A ∩ B)

n(A ∩ B) = 1000 – 800

n(A ∩ B) = 200

In the same question try finding people who can speak only English.

Hint: n(A U B’) = n(A-B) = n(A) – n(A ∩ B) will be applied here.

Solution: Let E represent English and H stands for Hindi.

So, n(E U H’) = n(E) – n(E ∩ H)

n(E U H’) = 450 – 200 = 250

Question: In a group of 50 persons, 25 like tea, 35 like coffee, and each person like atleast one of the two drinks. Solve:

  • How many like both tea and coffee?
  • How many like coffee only?
  • How many like tea but not coffee?

Solve and share your answer.

Solution credit: Harsh from Commerce batch (Shishu Niketan School, Mohali)

I hope it was helpful.

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