CSEC MATHEMATICS
Section 3 - Sets
A set is a clearly defined collection of objects/items which have common characteristics.
For example, the set of vowels of the English alphabet – {a, e, i, o, u} and, the set of multiples 5 less than 25 – {5, 10, 15, 20}.
We note the phrase ‘the set of’ is replaced by curly braces, { }.
Each item in a set is called an element or a member.
Symbols Meaning
is an element or member of
is a subset of or is contained in
is not a subset of or is not contained in
union – combine elements
intersection – refer to common elements
or { } null set or empty set
B1 or the complement of set B or all the
elements in the universal set that are not in set B
n(P) the number of elements or members in set P or the cardinal number of P Equivalent
= Equal
Exercise
1. 3 {1, 3, 5, 7} 2. 12 {1, 3, 5, 7, …, 19, 21} 3. – 1 W+
4. 1 is a prime number 5. {pink} {colours}
6. {k, l, o} {consonants}
7. 16 {x: x N and 16 x < 21}
8. {fork, spoon, plate} {rake, shovel, spade} 9. {0, 5, - 3, 7, 13} = {- 3, 0, 5, 7}
10. {4, 5, 6} {integers}
A finite set is a set in which it is possible to count or name all the elements in the set. For example,
M = {months of the year} or M = {January, February, March, April, …, November,
December}.
An infinite set is a set for which it is impossible to count or name all the elements in the set. For example, Y = {multiples of 7} or Y = {7, 14, 21, 28, 35, 42, …}.
The null or empty set contains no elements. For example,
H = {human beings on earth who are older than 300 years}.
Describe whether the following sets are finite, infinite or empty.
1. T = {desks in room 3}
2. P = {integers less than 5}
3. L = {natural numbers greater than 6 but less than 1}
4. R = {rational numbers less than 1}
5. H = {days of the week beginning with the letter T}
Two sets are said to be equal sets, if they both have the same elements. For example,
B ={6, 8, 27} and D ={8, 27, 6}.
Two sets are equivalent sets, if they both have the same ‘number of elements’. For example, G = {a, e, i, o, u} and P = {pink, red, blue, white, black}. That is n(G) = n(P) = 5.
Set Builder Notation
Set Builder Notation is a simple way of
Z = {6, 7, 8, 9, 10, 11, 12, …}is {x: x > 5, x Z}. This is read as ‘the set of all element of x such that x is greater than 5 and x is a member of the set of integers’.
Another example, is the set builder notation for the whole numbers between 1 and 5 is {x: 1 < x < 5}. This read as “the set of all element of x is greater than 1 but less than 5”, and is a
member of the set of whole numbers.
Exercise
Write the set that represents each of the following.
1. {x: x is a letter of the alphabet that follows d and comes before j}
3. {x: x Z and – 2 < x < 4}
4. the set of Political Parties in Jamaica. 5. {x: x W and – 3 < x < 7}
6. {x: x Z and – 2 ≤ x < 4} 7. {x: x W and – 3 ≤ x ≤ 7}
The members of a set can be described or listed, where possible. For example,
Describing the set – the set of vowels of the alphabet or {vowels of the alphabet}.
Listing the set – {a, e, i, o, u}.
Exercise
2. B = {prime numbers between 15 and 30} 3. C = {multiples of 5 between 12 and 47}
4. X = {whole numbers greater than or equal to 10 but less than 20}
5. Y = {letters used in the word ‘mathematics’}
Exercise
Describe each of the following sets. 1. P = {11, 13, 17, 19, 23}
2. R = {w, x, y, z}
3. S = {police, nurse, doctor, teacher}
4. M = {Toyota, ford, Nissan, audi, Honda} 5. T = {best fm, love101, rjr, power 106, irie fm}
If A and B are two sets and all the elements of A are contained in B, then we say that A is a subset of B. That is, A B. For example, H = {2, 9, 0, 6, 15} and K = {0, 6, 9}. That is, K H.
Exercise
Write or on the line to form a true statement.
1. P = {h, k, j, r} and W = {t, e, u, g} P ____ W
2. R = {11, 13, 17, 19, 21} and J = {prime numbers between 9 and 22}
J ______ R
3. D = {pink, red, purple, blue} and Q = {red, pink}
D _____ Q
H _____ L
5. P = {pencil, eraser, pen, sharpener} and G = {marker, crayon, ruler}
G ____ P
The total number of subsets of a set can be
calculated using, S = 2n, where n is the ‘number
of elements’ in the set. For example, the total number of subsets for L = {5, 7, 3} is S = 23 = 8,
since n(L) =3.
The subsets of L are {5}, {7}, {3}, {5,7}, {5, 3}, {7, 3}, {5, 7, 3} and { }.
NOTE The set itself and the empty set is a subset of any given set.
Write down all the subsets for each of the following sets.
1. V = {pen, ruler} 2. D = {5, 9, 6} 3. M = {l, y} 4. Z = {b}
5. B = {cup, plate, fork}
Exercise
Calculate the total subsets for each of the following sets.
1. V = {a, i, o, u, e}
2. A = {2, 4, 6, 8, 10, 12} 3. P = {pen, ruler, eraser, pencil}
4. Y = {first seven multiples of 8} 5. E = {factors of 24}
Exercise
Copy and complete the table below
Number of elements in a set(n)
Total
subsets (S)
Index
0 1 2 3 4 5 6
Venn Diagram/Universal Set
A Venn diagram can be represented using a
rectangle to represent the universal set and circle(s) inside the rectangle to represent a subset.
Fig. 1 shows a joint set. This Venn diagram is used when the two subsets have common
elements.
Fig. 2 shows a disjoint set. This Venn diagram is used when two sets do not have common
elements.
Fig. 3 shows that V J. This Venn diagram is used when all the elements of one set is a
member of another set.
Exercise
Use the information below to answer the questions which follow.
U
N F U A D U
J V
Given the universal set U = {11, 12,…, 17, 18} and two subsets, A = {numbers divisible by 3} and B = {odd numbers}, find:
1. A1 2. B1 3. A1 B1 4. A1 B1
5. A B 6. (A B)1 7. (A B)1
8. A only 9. B only 10. A B1
11. A1 B 12. n(B) 13. n(A1) 14. A B
15. Draw a Venn diagram to represent the information above.
Note: (A B)1 = A1 B1 (A B)1 = A1 B1
The Cardinal Number of a set G is the
8. n(B1) = 9. n(A1 B1) = 10. n((A B)1) =
Performing Set Operations
Find:
1. P1 given that U = {1, 2, 3, 4 , 5, 6, 7} and P =
{1, 3, 4, 7}
2. {7, 8, 9, 10, 11} {6, 8, 10, 12} 3. {1, 3, 5, 7, 9} {2, 4, 6, 8}
4. {1, 3, 5, 7, 9}
5. G1 given that U = {k, r, y, h, z, b} and
G = {r, h, z}
6. {7, 8, 9, 10, 11} {6, 8, 10, 12} 7. {1, 3, 5, 7, 9} {2, 4, 6, 8}
8. {1, 3, 5, 7, 9}
9. {l, m, k, t, o} {o, k, l}
Solving Problems involving Venn Diagrams NOTE: n(A B) = n(A) + n(B) – n(A B)
Exercise
1. Given n(A B) = 6, n(A) = 3 and n(A B) = 1, find n(B). Ans: 4
2. Given n(A B) = 25, n(B) = 17 and n(A B) = 7, find n(A). Ans: 15
3. Given n(P Q) = 28, n(P) = 19 and n(Q) = 17 find n(P Q). Ans: 8
4. Given n(D E) = 23, n(D) = 39 and n(E) = 37 find n(E D). Ans:53
A B1 means A only ; B A1 means B only
is associated with ‘and’ is associated with ‘or’
An equation in terms of x has an equal sign.
Exercise
Answer the following.
1. The Venn diagram below shows a survey of the days on which some students studied
over the weekend.
a. i. How many students were in the survey? ii. List the set of students in the survey. b. List the set of students who studied on: i. Saturday
ii. Sunday
iii. Saturday or Sunday iv. Saturday and Sunday v. Saturday and not Sunday vi. Sunday only
vii. neither Saturday nor Sunday.
2. The Venn diagram below shows the survey for some respondents in relation to their
preference of Reggae or Jazz music.
a. How many respondents were in the survey? b. How many respondents listened to:
i. reggae ii. jazz
iii. both jazz and reggae iv. reggae or jazz
v. jazz but not reggae vi. reggae only
vii. neither jazz nor reggae.
Exercise
U
20
Reggae Jazz
55 70
Answer the following.
1. In a class of 30 students, 20 played cricket, 17 played football and 7 played both cricket
and football. Each student played either cricket or football.
a. Draw a Venn diagram to represent the information above
b. How many students played:
i. cricket only ii. football only
2. In a class of 30 students, 21 like Mathematics, 12 like Physics and 6 like neither Mathematics nor Physics.
a. Draw a Venn diagram to represent the information above.
b. Determine the number of students who like: i. both Mathematics and Physics
iii. Physics only
3. In a community of 100 people, 31 like to run, 65 like to walk and 22 neither like to walk nor run.
a. Draw a Venn diagram to represent this information.
b. Determine the number of people who like: i. both to run and walk
ii. to run only. iii. to walk only.
4. The Venn diagram below shows the number of students who play the guitar (G) or the
violin (V) in a class of 40 students.
U
x
G V
2x 4
a. How many students play neither the guitar nor the violin?
b. Write an expression in terms of x, which represents the total number of students in the class.
c. Write an equation which may be used to determine the total number of students in the class.
d. How many students play the guitar? MAY 2015 Ques. 3a
15 families owned dogs 12 families owned cats
x families owned both dogs and cats 8 families owned neither dogs nor cats a. Given that U = {families in the survey}, C = {families who owned cats},
D ={families who owned dogs}, use the given information to complete the Venn diagram below.
b. Write an expression, in x, which represents the total number of families in the survey.
c. Write an equation which may be used to solve for x. JANUARY 2014 Ques. 3a
d. How many families owned dogs only?
U
C D
6. The universal set, U, is defined as the set of integers between 11 and 26. A and B are
subsets of U such that: A = {even numbers} and B = {multiples of 3}
a. How many members are in the universal set, U?
b. List the members of the subset A. c. List the members of the subset B.
d. Draw a Venn diagram to represent the relationships among A, B and U.
MAY 2014 Ques. 3a
French (F).
a. Represent this information on a Venn diagram.
b. Calculate the number of students who study Spanish (S) but not French (F).
c. Write, using set notation, the relationship between F and S. JANUARY 2014 Ques. 3a
8. A survey of 30 students in Form 5 showed that some students used cameras (C) or mobile phones (M) to take photographs.
20 students used mobile phones 4x students used only cameras
2 students did not use either cameras or mobile phones
a.Copy the Venn diagram below and complete it to show, in terms of x, the number of
students in each region.
b. Write an expression, in terms of x, which represents the total number of students in the survey.
c. Determine the number of students in Form 5 who used only cameras. MAY 2013 Ques. 3a d. How many students did not use a mobile phone?
U
M C
e. How many students did not use a camera phone?
9. There are 50 students in a class. Students in the class was given awards for Mathematics or Science.
36 students received awards in either Mathematics or Science.
6 students received awards in both Mathematics and Science.
2x students received awards for Mathematics only.
x students received awards for Science only
M = {students who received awards for Mathematics},
S = {students who received awards for Science}
a. Copy and complete the Venn diagram to
represent the information about the awards given, showing the number of students in each subset. b. Write an expression in terms of x to represent the number of students in the class.
c. Calculate the value of x. 10 students JANUARY 2013 Ques. 3a
d. How many students did not receive an award in Mathematics or Science? 14 students
U
M S
10. In a survey of 36 students, it was found that 30 play tennis
x play volleyball only
9x play both tennis and volleyball 4 play neither tennis nor volleyball
a. Given that U = {students in the survey}, V = {students who play Volleyball},
T = {students who play Tennis}
Copy and complete the Venn diagram below to show the number of students in the subsets
marked y and z. U
9x
V T
x y
b. Write an expression in x to represent the total number of students in the survey.
c. Write an equation in x to represent the total number of students in the survey and hence solve for x. MAY 2012 Ques. 3a
11. The Venn Diagram below represents information on the types of games played by members of a youth club. All members of the club play at least one game.
S represents the set of members who play squash.
U
Mia
T H
Neil S
T represents the set of members who play tennis. H represents the set of members who play
hockey.
Leo, Mia and Neil are three members of the youth club.
a. State what game(s) is/are played by: i. Leo
ii. Mia iii. Neil
b. Describe in words the members of the set H1
12 a. Describe, using set notation only, the
shaded region in each Venn diagram below. The first one is done for you.
A B i. _________ ii. ________
iii. __________ iv. ___________
12. Draw a Venn diagram and shade the region which represents each of the following set
notation.
a. B1 b. (A B)1 c. B1 A
d. B1 A1 U
A B U
A B
U
A B
U
A B
U
