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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Venn Diagram and Subset Relation
MATH0201
BASIC CALCULUS
Set Operations
Set
Sample Space
Dr. WONG Chi Wing
Department of Mathematics, HKU
General Sets
Set Builder Notation
Some Examples
MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Definition 1
A set is a well–defined collection of objects, i.e., we can
determine whether a given object belongs to that set or not
without any ambiguity.
We may represent a set by
1. WORD DESCRIPTION
S is the set of the first five prime numbers.
Example 2
2. LISTING
S := {2, 3, 5, 7, 11}.
Which of the following is/are a set/sets?
1. The collection of HSI constituent stocks.
2. The collection of HSI constituent stocks with attractive
dividend.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Sets are usually denoted by capital letters: A, B, C, S, T etc.
Example 3
Definition 4
The following two sets are indeed the same:
A member of a set is called an element of the set.
1. The set of solutions of x 2 − 1 = 0.
If x is an element of a set S, then we write x ∈ S.
2. {−1, 1}.
If x is NOT an element of S, then we write x ∈
/ S.
This set is also called the solution set of the equation
x 2 − 1 = 0.
Example 5
2 ∈ {2, 3, 5, 7, 11} but 1 ∈
/ {2, 3, 5, 7, 11}.
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MATH0201 BASIC CALCULUS
Venn Diagram and Subset Relation
We may represent a set graphically by using a Venn Diagram.
Definition 6
Example 8
A set containing no element is called an empty set, which is
denoted by ∅.
We may represent the sets U := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and
A := {1, 3, 5, 7, 9} by Venn diagrams.
Example 7
We may combine them together
Which of the following is an empty set?
(a) Solution set of 3x 2 + 2 = 0.
(b) Solution set of x 2 + 2x + 1 = 0.
(c) {∅}.
Every element of A is an element of U.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Venn Diagram and Subset Relation
Venn Diagram and Subset Relation
Example 9
Definition 10
We may represent the sets U := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and
B := {1, 3, 5, 7, 9, 11} by Venn diagrams.
Let S be a set. A subset E of S is a set with the property that
every element of E is an element of S. When this is the case
we write E ⊆ S.
If E is not a subset of S, then we write E 6⊆ S.
Example 11
{1, 3, 5, 7, 9} ⊆ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
{1, 3, 5, 7, 9, 11} 6⊆ {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
2 ∈ U but 2 ∈
/ B.
11 ∈ B but 11 ∈
/ U.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Venn Diagram and Subset Relation
Set Operations
Definition 14 (Union)
Remark 1
∅ is a subset of any set.
Let A and B be sets. Then the union of A and B, denoted by
A ∪ B, is the set of all the elements of A and B.
Definition 12
Example 15
Two sets A and B are equal if every element of A is an element
of B and vice versa. We shall write A = B.
{1, x} ∪ {x, C} = {1, x, C}.
Example 13
Definition 16 (Intersection)
{−1} = solution set of x + 1 = 0.
Let A and B be sets. Then intersection of A and B, denoted by
A ∩ B, is the set of all the common elements of A and B.
{−3, −1} = solution set of x 2 + 4x + 3 = 0.
Example 17
Clearly, A = B if and only if A ⊆ B and B ⊆ A.
{1} ∩ {x, C} = ∅. {1, x} ∩ {x, C} = {x}.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Set Operations
Set Operations
Definition 18
Definition 21
The set of all the objects to be considered is called a universal
set.
Let A and B be sets. Then the (relative) complement of A in B,
denoted by B \ A, is the set of the elements in B which is NOT
in A.
Definition 19
Let U be a universal set and A ⊆ U. The complement of A,
denoted by Ac , is the set of all the elements of U which is not
an element of A.
Example 22
1. {2, 3, a, c} \ {1, 2, 3} =
Example 20
2. {1} \ {x, C} =
If U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {1, 3, 5, 7, 9}, then
3. {1} \ {1, 2, 3} =
Ac = {2, 4, 6, 8, 10} .
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Set Operations
Sample Space
Theorem 23
Let A and B be subsets of a universal set. Then
1. A = (A ∩ B) ∪ (A \ B).
Inherent in any discussion of probability is the performance of
an experiment in which a particular result, or outcome, involves
chance.
2. (de Morgan’s Laws)
For example, when tossing a coin (the
experiment), the outcome is either a
head (H) or a tail (T), but the actual
outcome is determined by chance.
(A ∪ B)c = Ac ∩ B c and (A ∩ B)c = Ac ∪ B c .
=
T
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Sample Space
Sample Space
Definition 26
Definition 24
An event E for an experiment is a subset of the sample space
for the experiment.
A sample space S for an experiment is the set of all possible
outcomes of the experiment.
The elements of S are called sample points.
Example 27
Example 25
Roll a die and observe the number on the
top face.
The sample space for this experiement is
The sample space of tossing a coin is S = {H, T }.
{1, 2, 3, 4, 5, 6}.
Now two distinct coins are tossed and the result for each coin is
observed. The sample space of this experiment is
{HH, HT , TH, TT }; where HH means both coins show head,
HT means the first coin shows a head while the second coin
shows a tail, and so on.
1. {1} is the event of showing one.
2. {1, 3, 5} is the event of showing odd
numbers.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
Sample Space
General Sets
Example 28
Definition 29
Continue with Example 27. Let E1 be the event of showing odd
numbers and E2 be the event that showing a number greater
than 2.
E1 = {1, 3, 5} and E2 = {3, 4, 5, 6}.
A set is said to be finite if it has finitely many elements.
Otherwise, it is said to be infinite.
Example 30
These are finite sets.
What are the following events?
1. {2, 4, 7} as it has 3 elements.
1. Either an odd number is shown or the number is greater
than 2;
2. ∅.
2. an odd number greater than 2 is shown;
These are infinite sets.
3. an even number is shown;
1. The set of all odd numbers.
4. an even number greater than 2 is shown.
2. The solution set of x − 3 < 0.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
General Sets
General Sets
Set Builder Notation
Set Builder Notation
Example 31
We may represent an infinite set by
{x : x is one of the first five odd number} = {1, 3, 5, 7, 9}.
1. WORD DESCRIPTION
The set of all the odd numbers.
Example 32
2. LISTING
{1, 3, 5, 7, 9, . . .}.
The following three sets are indeed the same:
1. The set of solutions of x 2 − 1 = 0.
3. SET–BUILDER NOTATION
2. {−1, 1}.
3. {x : x 2 − 1 = 0}.
{ x : x is an odd number }.
We may also write
It is read as the set of all elements x such that x is an odd
number.
{−1, 1} = {x : x 2 − 1 = 0}.
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MATH0201 BASIC CALCULUS
General Sets
General Sets
Set Builder Notation
Some Examples
Example 34 (Set of NUMBERS)
Example 33
I
The union, intersection, and complement of sets can be defined
by
N is the set of all natural numbers
N := {1, 2, 3, 4, . . .}.
1. A ∪ B = {x : x ∈ A or x ∈ B}.
I
2. A ∩ B = {x : x ∈ A and x ∈ B}.
3. Ac = {x : x ∈ U but x ∈
/ A}.
Z is the set of all integers
Z := {. . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . .}.
4. A \ B = {x : x ∈ A but x ∈
/ B}.
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Q is the set of all rational numbers.
I
R is the set of all real numbers.
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MATH0201 BASIC CALCULUS
MATH0201 BASIC CALCULUS
General Sets
General Sets
Some Examples
Some Examples
Example 36
Example 35 (Intervals)
Refer to the graph y = x 2 − 2x − 3 below,
Let a, b ∈ R and a < b.
1. (a, ∞) := {x ∈ R : a < x}.
2. (−∞, a) := {x ∈ R : x < a}.
3. [a, ∞) := {x ∈ R : a ≤ x}.
4. (a, b) := {x ∈ R : a < x < b}.
Similarly, we have (−∞, a], [a, b), (a, b] and [a, b].
use the listing method or interval notation to represent the
solution set of x 2 − 2x − 3 > 0.
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MATH0201 BASIC CALCULUS
General Sets
Some Examples
Example 37
Find the solution set of the compounded inequality
x −1<2
−2 < x − 1.
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