How To Solve Sets With Venn Diagram

Are you finding it difficult to understand and solve Sets with Venn diagram?

In this lesson,

You will learn the step-by-step method for solving questions on Sets with Venn diagrams in your mathematics exam.

ALSO SEE: How To Identify and Interpret Basic Set Symbols

Take note.

A link for you to practice past exam questions on sets with Venn diagrams is included at the bottom of this lesson.

Solving sets with Venn diagram

Let’s take an example.


50 students were asked what they did last night. 16 said they read a book, and 41 said they watched television. If 7 said they did neither, how many did both?

1. List the details in the question

  • Total number of students is the Universal Set (\mho) = 50
  • Number of students who read a book is subset (B) = 16
  • Number of students who watch Television is subset (T) = 41
  • Number of students who neither read a book (B’) nor watch Television (T’) is therefore, B’ \cup T’ or (B \cup T)’ = 7
  • Number of students who both read a book and watch television (B \cap T) = ?? Let’s represent the unknown with (x)

2. Transfer the details into a Venn diagram.


(x) is being subtracted from the subset B and T because what they both have in common (x) must be removed to get what they each have alone.

3. Convert your Venn diagram to an equation.

Add all the elements in your Venn diagram and equate them to the Universal Set.

In other words,

(Subset B) + (Intersection) + (Subset T) + (Complement) = (Universal Set)

  • 16 – x + x + 41 – x + 7 = 50

4. Solve for x in the equation

16 – x + x + 41 – x + 7 = 50

  • (-x) cancels (+x)

: 16 + 41 – x + 7 = 50

  • Addition

: 57 – x + 7 = 50

  • Collect like terms

: 57 – 50 + 7 = x

  • Addition before subtraction

: 57 – 43 = x

14 = x


Number of students who both read a book and watch television (B \cap T) represented as (x)

= 14

Test yourself

Click on the link below to practice past questions on Sets from different Mathematics exam.


Do you have something to ask on Sets with Venn diagram? Comment below and a staff or fellow ACADite will reply you.

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