Venn Diagrams with Three Sets (Edexcel IGCSE Maths)

Spec point — 1.5E

Venn diagrams with three sets

What does a Venn diagram with three sets look like?

• There is a rectangle representing the universal set

• There are three circles

  • One for each of the sets

    • E.g. $A$, $B$ and $C$

• The three circles intersect and split the rectangle into eight regions

  • A region where all three circles intersect

    • $A\cap B\cap C$

  • Three regions where exactly two circles intersect

    • $A\cap B\cap C'$, $A\cap B'\cap C$ and $A'\cap B\cap C$

  • Three regions where each circle does not intersect with any other circle

    • $A\cap B'\cap C'$, $A'\cap B\cap C'$ and $A'\cap B'\cap C$

  • A region outside the three circles

    • $A'\cap B'\cap C'$
    • This can also be written as $( A \cup B \cup C ) '$

The intersections of three sets

Example of a Venn diagram with three sets

• It is possible that two of the circles do not intersect

  • E.g. if $A\cap C=\emptyset$

Example of a Venn diagram where two sets do not have an intersection

How do I find the number of elements in a subset?

• Identify the intersections which make up the subset

  • E.g. the subset $A\cap B$ is made up of $A\cap B\cap C$ and $A\cap B\cap C'$

• Add together the number of elements in the intersections

Example of finding number of elements in subsets

How do I fill in a Venn diagram with three sets?

• Start with the intersection of all three circles $A\cap B\cap C$

  • Fill in the number or label it $x$ if it is unknown

• Fill in regions where exactly two circles intersect

  • You might be given the total number of elements in the intersection between those two sets

    • E.g. There are 20 elements that are in both set $A$ and set $C$

  • Subtract the number in the intersection of all three circles to find the number of elements that are just in those two sets

    • E.g. $20-x$ elements are in set $A$ and set $C$ but not set $B$

• Fill in the parts of the circles which do not intersect other circles

  • You might be given the total number of elements in a set

    • E.g. There are 60 elements in set $A$

  • Subtract the numbers in the intersections which involve set $A$

    • E.g. subtract the number of elements in $A\cap B\cap C$, $A\cap B\cap C'$ and $A\cap B'\cap C$ from the number of elements in $A$

• Fill in the number outside all the circles

  • This is the total number of elements minus the number of elements in all the intersections

Worked Example Some students were asked whether they like studying statistics $( S )$, algebra $( A )$ and geometry $( G )$.

• 5 said they like studying all three of statistics, algebra and geometry
• 11 said they like studying statistics and algebra
• 16 said they like studying algebra and geometry
• 8 said they like studying statistics and geometry
• 25 said they like studying geometry
• 4 said they do not like studying any of the three topics
• the number who said they like studying statistics only is the same as the number who said they like studying algebra only

Let $x$ be the number of students who said they like studying statistics.

(a) Show all this information on the Venn diagram, giving the number of students in each appropriate subset, in terms of $x$ where necessary. Simplify all expressions.

Answer:

Put 4 outside the circles

Put 5 in the intersection of all three circles

Find the number of students who like statistics and algebra only

11 - 5 = 6

Find the number of students who like algebra and geometry only

16 - 5 = 11

Find the number of students who like statistics and geometry only

8 - 5 = 3

Find the number of students who like geometry only

25 - 5 - 11 - 3 = 6

Find the number of students who like studying statistics only

• Give your answer in terms of $x$

$x-5-6-3=x-14$

The number of students who like algebra only is the same as the number of students who like statistics only

(b) Given that 30 students said they like studying algebra. Find the number of students who said they like studying statistics.

Answer:

Add together the numbers in the circle for algebra

$x-14+6+5+11=x+8$

Set this equal to 30 and solve for $x$

$\begin{gathered} x + 8 & = 30 \\ x = 22 \end{gathered}$

Final answer: 22 students said they like studying statistics