WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
1 | P a g e
Definition of a Set
A set is a collection of well defined objects.
Capital letters are used to denote a set and set brackets {} are used to list the objects that can be
found in that set.
Building a Set
A set must be defined.
Ex. Let A be the set of the first 5 even numbers.
Next find the objects that belong to this set.
Ex. 2, 4, 6 ,8 ,10
State the set.
Ex. A = {2, 4, 6, 8, 10}
Common Sets
The set of natural numbers or counting numbers: N = {1, 2, 3, 4, 5, …..}
The set of whole numbers: W = {0, 1, 2, 3, 4, 5, …..}
The set of integers: Z = {….., -3, -2, -1, 0, 1, 2, 3,…..}
1. Let M be the set of the first 7 prime numbers. Using set notation, state the set M.
M = { }
Even numbers are
numbers that can be
divided by 2 without
leaving a remainder.
There are 5 objects in
ascending order in this
set and all the objects
are divisible by 2
A prime number is a
number that can be
divided by 1 and itself.
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
2 | P a g e
2. Let P be the set of 5 common flavours of ice-cream. Using set notation, state the set P.
P = { }
3. Let K be the set of 5 popular brands of cars in Trinidad. Using set notation, state the set
K.
K={ }
4. Let G be the set of the first 10 odd numbers.
Using set notation, state the set G.
G = { }
Belonging to a Set
An object that belongs to a set can be represented using the symbol ‘∈’ otherwise if it does not
belong to the set, the symbol ‘∉’ is used.
Let A be defined as follows: A = {2, 4, 6, 8, 10}
Determine which of the numbers from 1 through 10 belongs to the set A.
By inspection,
2 ∈A,
4 ∈A,
6 ∈A,
8 ∈A,
10 ∈A
1∉A
3 ∉A
5 ∉A
7 ∉A
9 ∉A
An odd number is a
number that leaves a
remainder of 1 when
divided by 2
These elements
can be found in A
These elements cannot
be found in A
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
3 | P a g e
The Empty or Null Set
Consider an empty school bag or an empty wallet or an empty room. If one was to look for a
book or money or a person respectively, nothing will be found.
The empty set, therefore, is the set that contains no objects and is given the set symbol, {∅}.
N.B. The empty set is a subset of all sets.
Let A be the set of pigs that fly. Using set notation, state the objects that belong to A.
Since there are no pigs that fly, A = {∅}.
Let D be the set of all dragons that could fly. Using set notation, state the objects that belong to
D.
Since there are no dragons that fly, D = {∅}.
Finite Set
A finite set is a set where there is a constriction on the number of objects that can be placed in a
set.
Ex. Let B be the set of the first 4 multiples of 5.
B = { }
The
constriction
here is the
“first 4”
Multiples are what you get
after multiplying a number
by a positive integer.
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
4 | P a g e
Infinite Set
An infinite set is a set that has no restrictions on what can be placed in a set.
Examples of infinite sets can be identified by the use of the three dots (...) and means to continue
on.
N = {1, 2, 3, 4, 5,…}
W = (0 , 1, 2, 3, 4, 5,…}
The Universal Set
The universal set, denoted by U, is the set that contains all objects or everything.
It is useful to note that the word ‘everything’ refers to what is relevant to the question.
All sets are taken from the universal set.
Ex. Let U be the set of natural numbers.
U = {1, 2, 3, 4, 5 , 6, 7, 8, 9, 10,…}
Sets can be built from the set U:
The set of even numbers between 1 and 20 – { }
The set of the first 10 prime numbers – { }
The set of odd numbers between 50 and 60 – { }
The set of the first 8 multiples of 5 – { }
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
5 | P a g e
Subsets
A subset of any set U is the set that contains all or some of the objects that can be found in U,
where U is the universal set.
Alternatively, Let A & B be two sets, A is a subset of B if all the objects in A can be found in B.
This is denoted by, A ⊆ 𝐵 otherwise if no elements in A can be found in B, then this is denoted
by A ⊈ 𝐵.
Proper Subsets
Let A & B be two sets, A is a proper subset of B if the objects in A can be found in B. However
not all the objects in B can be found in A.
This is denoted by, A ⊂ 𝐵 otherwise if no elements in B can be found in A, then this is denoted
by A ⊄ 𝐵.
The following example will illustrate.
Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?
1 ∈ A, and 1 ∈ B as well.
3 ∈ A and 3 ∈ B.
4 ∈ A, and 4 ∈ B.
That's all the elements of A, and every single one is in B.
Yes, A is a subset of B ie 𝐴 ⊂ 𝐵
NB. Although 2 ∈ B, it cannot be found in A, ie 2∉ 𝐴
To determine the number of subsets that can be formed from any given set, the following
formula is useful:
2𝑛
n represents the
number of elements in
the set.
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
6 | P a g e
Determine and list the number of subsets that can be form from the set X, where X = {a, b, c}
Since X has 3 elements, the number of subsets that can be formed is 23 = 2 × 2 × 2 = 8 subsets
The subsets are:
No elements: {∅}
One element: {a}, {b}, {c}
Two elements: {a, b}, {a, c}, {b, c}
Three Elements: {a, b, c}
The subsets are: {{∅}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}
Points to note:
First use the formula to determine the number of subsets.
The empty set and the set itself are subsets.
These two sets will always be subsets of any given set.
Always form subsets in a systematic way starting with the set that has zero objects followed by
the set with one object followed by the set with two objects followed by the set with 3 objects
and so on.
Equal Sets
Two sets are said to be equal if and only if they contain the same objects. The order in which the
objects are placed does not matter.
Ex. A = {1, 2, 3, 4} B = {2, 1, 4, 3}
All the elements in A are in B and all the elements in B are also in A.
Since 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴, then A = B
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
7 | P a g e
It is observed by inspection that A is a subset of B and B is a subset of A.
Since both are subsets of each other, they are equal.
Therefore a necessary condition for two sets to be equal is they have to be subsets of each other.
Cardinality
The cardinality of a set represents the number of objects that can be found in a set by counting.
Let Z be the set of vowels.
Z = { a, e, i, o, u}
The cardinality of Z, written as n(Z) = 5.
Let T be the set of constants. Using set notation, state the set T and hence find the cardinality of
T.
T = { }
n(T) =
Equivalent Sets
Two sets are said to be equivalent if their cardinality are equal.
Ex. M = {1, 2, 3, 4} N = {a, b, c, d}
n(M)=4 n(N)=4
Since the cardinality of M and N are equal, M & N are equivalent sets.
n(Z) means the
number of
elements in Z
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
8 | P a g e
Compliment of a Set
Let A be a set. The compliment of the set A written as 𝐴’ represents all the objects that are not in
A.
Ex. Let U be the first 10 counting numbers.
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Let A be the first 5 even numbers.
A = {2, 4, 6, 8, 10}
Therefore A’ = {1, 3, 5, 7, 9}
Union
Consider the union of a bride and groom. The bride represents one set and the groom represents
another set. At the marriage ceremony, both the bride and groom come together with their
respective families.
In set theory, the union of two sets is just this.
If A and B are two sets, the union of A and B written as 𝑨 ∪ 𝑩 is the set that contains all the
elements found in A and B.
Let A = {1, 2, 3, 4} and let B = {a, b, c, d}
The union of A and B, 𝑨 ∪ 𝑩 = {𝟏 , 𝟐, 𝟑, 𝟒, 𝑎, 𝑏, 𝑐, 𝑑}
The objects 1, 2,
3, 4 are found in
the set A
The objects a, b,
c, d are found in
the set B
𝐴 ∪ 𝐵 represents the
combination of
elements belonging to
both A and B
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
9 | P a g e
1. Let X = {2, 4, 6, 8, 10} and Y = {1, 3, 5, 7, 9}. Using set notation, state:
a. The cardinality of X
b. The cardinality of Y
c. The union of X and Y
d. If X and Y are equivalent or equal
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
10 | P a g e
Intersection
Consider a boy and girl who wants to be with each other in a relationship. The boy will have his
likes and the girl will have her likes. The common things they both like together would represent
the intersection of their likes.
Therefore the intersection of two sets can be stated as:
If A and B are two sets, the intersection of A and B written as 𝐴 ∩ 𝐵, represents the objects that
are common to both A and B, ie, if it can be found in A, it can be found in B and if it can be
found in B, it can be found in A.
Let A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5, 6}
The intersection of A and B, 𝐴 ∩ 𝐵 = { }
The objects
2, 4, 6 can be
found in the
set A
The objects 2, 4, 6 are
common to both A and B,
therefore they will be found
in the intersection
The objects 2, 4,
6 can be found in
the set B
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
11 | P a g e
1. Let X = { 5, 10, 15, 20} and Y = {10, 20, 30}. Using set notation, state:
a. The cardinality of X
b. The cardinality of Y
c. The union of X and Y
d. The intersection of X and Y
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
12 | P a g e
Venn Diagrams
A Venn diagram is a pictorial representation of sets. A rectangle is used to represent the
universe, U and circles are used to represents the sets themselves.
1. In the Venn diagram the shaded circle represents the objects that belong to A and the
unshaded region represents the objects that do not belong to A.
2. The shaded region represents the objects that belong to A and also belong to B, ie, A is
contained in B. A is a subset of B or 𝐴 ⊂ 𝐵 .
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
13 | P a g e
3. The shaded region represents the intersection or 𝐴 ∩ 𝐵.
4. The shaded region represents the union of A and B or 𝐴 ∪ 𝐵.
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
14 | P a g e
5. The shaded region represents the objects that belong to A only or 𝐴 ∩ 𝐵′
6. The shaded region represents the objects that belong to B only or 𝐴′ ∩ 𝐵
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
15 | P a g e
Consider two parties that have the following refreshments:
Party A = {Chips, Cookies, Cake, Soft Drink, Juice}
Party B = {Soft Drink, Cutters, Chips, Peanuts}
Both Party A & Party B have something in common;
A ∩ 𝐵 = {Chips, Soft Drink}
If we were to determine how was Party A unique, such that Party B does not have the objects in
Party A, this would just be the set: {Cookies, Cake, Juice}
It is important to note that these objects that are found in A AND not in B.
Using set notation, this translates to A ∩ 𝐵′.
Likewise, Party B is unique in such a way such that Party A does not have the objects in party B,
this is the set {Cutters, Peanuts}.
These are the objects that are found in B AND are not in A.
Using set notation, this translates to B ∩ 𝐴′.
A ∩ 𝐵′ = { }
𝐵 ∩ 𝐴′ = { }
WOODBROOK SECONDARY SCHOOL
MATHEMATICS
SET THEORY
FORM 4
16 | P a g e
Determine the cardinality of:
The set A n(A) =
The set B n(B) =
The set A ∩ 𝐵 n(A ∩ 𝐵) =
The set A ∩ 𝐵′ n(A ∩ 𝐵′) =
The set B ∩ 𝐴′ n(B ∩ 𝐴′) =
Next find 𝐴 ∪ 𝐵 = {Chips, Cookies, Soft Drink, Cutters, Peanuts, Juice, Cake}
n(𝐴 ∪ 𝐵) =
Note that n(A) + n(B) = 5 + 4 = 9
Therefore, n(A ∩ 𝐵) + n(A ∩ 𝐵′) + n(B ∩ 𝐴′) = 2 + 3 + 2 = 7
Points to note:
n(A ∩ 𝐵′) = n(A) - n(A ∩ 𝐵) = 5 – 2 = 3
n(B ∩ 𝐴′) = n(B) - n(A ∩ 𝐵) = 4 – 2 = 2
A general rule to follow is given as follows:
𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵)
Using the example before:
𝑛(𝐴 ∪ 𝐵) = 5 + 4 – 2 = 9 – 2 = 7
𝑛(𝐴 ∪ 𝐵) = 7
This is wrong since the total number of
objects in A and B is 7 (two objects are
repeated)
This represents the total number
of objects put together
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