1. Venn Diagrams

2. Venn diagrams are used to represent sets of numbers or objects or things.
The universal set is usually represented by a rectangle and the sets within it are usually represented by circles or ellipses.
Here is an example of the universal set with set A and A’ shaded.

3. An example:
Given that U = {2, 3, 5, 7, 8} and A = {2, 7, 8} draw the Venn Diagram, including listing in the diagram where the numbers in each set lie, and shade A’.

4. Subsets: If , then every element of B is contained within A as every element of B is also in A. This is shown as B drawn entirely inside A in the diagram.
Intersection: The region contains all elements common to both A and B. This is shown where A and B overlap in the diagram.

5. Union: The region consists of all elements in A or B
Union: The region consists of all elements in A or B. This is shown as all of A or B in the diagram.
Disjoint (or Mutually Exclusive) Sets: Disjoint sets don’t have any common elements where This is shown as two circles which do not intersect.

6. Given that U = {1, 2, 3, 4, 5, 6, 7, 8} illustrate on a Venn Diagram the sets A = {1, 3, 6, 8} and B = {2, 3, 4, 5, 8}.
Given that U = {1, 2, 3, 4, 5, 6, 7, 8} illustrate on a Venn Diagram the sets A = {1, 3, 6, 7, 8} and B = {3, 6, 8}.

7. Given that U = {1, 2, 3, 4, 5, 6, 7, 8, 9} illustrate on a Venn Diagram the sets A = {2, 4, 8} and B = {1, 3, 5, 9}.

8. Given two overlapping sets A and B, there are four regions on a Venn Diagram which are noted.
is in both A and B.
is in A but not B.
is in B but not A.
is neither in A nor in B.

9. We can shade various regions that are being considered.
For example, for two intersecting sets, we have the following diagram.
Shade set A.

10. shade
shade B’

11. shade
shade

12. shade
shade

13. Without looking at your notes, copy the following Venn Diagram and shade the region that represents the following:
(one Venn Diagram for each question!)
a) A
b) A’ c) d) e) f)

14. Given U = {apple, pear, banana, pineapple, watermelon, orange, mango}, P = {apple, pear, banana, pineapple} and Q = {apple, pear, orange}. Draw the corresponding Venn Diagram and find the number of elements in the following:
a) P
b) Q’ c)
d) P, but not Q
e) Q, but not P
f) neither P nor Q
Answers:
a) n(P) = 4
n(Q’) = 4 c)
n(P, but not Q) = 2
n(Q, but not P) = 1
f) n(neither P nor Q) = 2

15. Given the following Venn Diagram and if (3) means that there are 3 elements in the set , how many elements are there in the following sets?
Answers:
a) n(P) = 10
n(Q’) = 11 c)
n(P, but not Q) = 7
n(Q, but not P) = 11
f) n(neither P nor Q) = 4
a) P
b) Q’ c)
d) P, but not Q
e) Q, but not P
f) neither P nor Q

16. Given the following Venn Diagram, State the number of elements in:X Y’ c)
X, but not Y
Y, but not X
f) neither X nor Y
Answer: 13
b) 16
c) 24
d) 7
e) 11
f) 9

17. Given that n(U) = 30, n(A) = 14, n(B) = 17 and , find:
Answer: 25
b) 8
Given that n(U) = 50, n(A) = 33, n(B) = 27 and , find: a) b)
Answer: 42
b) 9

18. Given that n(U) = 25, n(A) = 10, n(B) = 12 and , find:
Answer: 8
b) 5
Given that n(U) = 55, n(A) = 34, n(B) = 27 and , find: a) b)
Answer: 45
b) 11

19. A Swedish curling club has 32 members
A Swedish curling club has 32 members. 22 have blonde hair, 24 have blue eyes, and 15 have both blonde hair and blue eyes. Find the number of Swedish curlers who have:
a) blonde hair or blue eyes
b) blonde hair, but not blue eyes
c) neither blonde hair, nor blue eyes.
Answer: 31
b) 7
c) 1
A squash club has 27 members. 19 have black hair, 14 have brown eyes and 11 have both black hair and brown eyes.
Draw the corresponding Venn Diagram.
Find the number of members with black hair or brown eyes.
c) Find the number of members with black hair, but not brown eyes.
Answer: a) 22
c) 8

20. In an athletics club with 20 members, 15 compete in the 100m, 12 compete in the 400m and 9 compete in both. Find the number of athletes who:
a) compete in the 100 or 400m
b) compete in the 100m, but not the 400m
c) neither compete in the 100m nor the 400m
Answer: 18
b) 6
c) 2
Out of a class of 30 students, 22 study math, 15 study Biology and 7 study both. Find the number of students who:
a) study math or Biology
b) study Biology but not math
c) neither study math nor Biology
Answer: 30
b) 8
c) 0

21. When we have two sets A and B:
Notice that A and B overlap in the intersection and, when this happens we count it twice. So, we must subtract one of these.
So, we get the formula:

22. A platform diving squad of 25 has 18 members who dive from 10m and 17 who dive from 4m. How many members dive from both platforms?
Answers:
10 members
A group of 30 students were surveyed about dental hygiene. 16 students floss, 15 use mouthwash, and 3 use neither. How many use both?
Answers:
4 students

23. In a school of 500 students, 260 play hockey and 220 play tennis
In a school of 500 students, 260 play hockey and 220 play tennis. If 400 students play either hockey or tennis, how many students play:
both sports?
b) neither sport?
Answer: 80
b) 100
In a class of 40 students, 23 have dark hair, 18 have brown eyes and 26 have dark hair, brown eyes or both. Determine the number of students who:
have dark hair and brown eyes.
b) dark hair but not brown eyes.
Answer: 15
b) 8

24. The number of elements in the sets A and B are shown in the Venn Diagram below. If , find the value of p.
Answer:
p = 15

25. Of 450 interviewed, 135 contributed to Red House, 180 contributed to Blue House and 180 contributed to Yellow House. 27 contributed to Red House and Yellow House, 54 contributed to Yellow House and Blue House while 45 contributed to Red House and Blue House. 63 of the people interviewed contributed to none of the houses. How many contributed to all three houses?
Answer:
18 people