2. Chapter 1
Basic Set Theory
1.1 Basic Set Definitions
1.1.1 Definition
A set is an unordered collection of objects.
the objects in a set is called elements of this set.
Note that since a set is unordered, then two sets A, B such that A =
{1, 2, 3} and B = {3, 2, 1} satisfy that A = B, or in other word, they are
”identical”.
The point is: changing the sequence of elements within a set will
not create a new set.
We write a ∈ A to denote that a is an element of the set A. The notation
a ∈ A denotes that a is not an element of the set A.
1.1.2 How to Define sets
Since we know the definition of sets, then the question comes: how to define
sets using mathematical language?
There are several ways.
1. roster method
Idea: list all the members of a set.
For example, A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}, a set of all natural numbers
that are less than 10.
2. set builder
Idea: characterize all those elements in the set by stating the
property or properties they must have to be members.
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3. For example,A = {x | x is an even interger that is greater than 0}
1.1.3 useful sets
Here are some very useful sets:
N = {0, 1, 2, 3...}, the set of natural numbers
Z = {..., −2, −1, 0, 1, 2, ...}, the set of integers
Z+
= {1,2,3,...}, the set of positive integers
Q = {p/q | p ∈ Z, q ∈ Z, q = 0}, the set of rational numbers
R, the set of real numbers
R+
, the set of positive real numbers
C, the set of complex numbers
1.1.4 Intervals
When a, b are real numbers such that a < b, we have:
[a, b] = {x | a ≤ x ≤ b}
(a, b] = {x | a < x ≤ b}
[a, b) = {x | a ≤ x < b}
(a, b) = {x | a < x < b}
1.1.5 When sets are equal
Definition: Two sets A and B are equal, if and only if they have the same
elements. Therefore:
• For every a ∈ A, a ∈ B as well
• For every b ∈ B, b ∈ A as well
We write A = B if the two sets A and B are equal.
1.1.6 empty sets
There is a special set that has no elements.
This set is called the empty set, or null set, and is denoted by ∅. There
is an alterative way to write an empty set as { }, a pair of braces within
nothing inside.
1.1.7 Venn diagrams
This diagram gives us a more intuitive way understand sets, especially the
relations between different sets.
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4. 1.1.8 Subsets
We are very interested to see the relations between different sets, so we
introduce subset of a set as:
Definition: The set A is a subset of B if and only if every element
of A is also an element of B.
If A is a subset of B, we write A ⊆ B.
Question: how to determine that one set, say A, is a subset of another,
say B?
• A is an empty set
In this situation, A is clearly a subset of B, regardless of what B might
be.
• A is not empty but B is empty
In this case, A is not a subset of B.
• Neither A nor B is empty
check whether every x belonging to A also belongs to B.If this
is true, then A is a subset of B. If not, then A is not a subset of B.
Note that:
by saying A is a subset of B, it is possible that A is actually equal to B.
Question: what if a set A is truly a ”subset” of another set B?
Definition: a set A is a proper subset of a set B if A is a subset of B
and A = B.
We write A ⊂ B to show that A is a proper subset of B.
Similarly, to check whether a set A is a proper subset of another set B,
we can follow the patterns of determining whether A is a subset of B, but
we have to do more:
• A is an empty set
Given this condition, we know that A is a subset of B. We go a step
forward:
If B is not empty, then A is a proper subset of B, as A = B.
However,if B is empty as well, then A = B, which means that A is
NOT a proper subset of B.
We can generalize this idea like this:
An empty set is a proper subset of any other nonempty set.
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5. • A is not empty, B is empty
Under this condition, A is not a proper subset of B as A is not a subset
of B.
• Neither A nor B is empty
Check whether every element belonging to A also belongs to B. If this
is false, then A is not a proper subset of B. If this is true, we have to
do another check:
Check whether there is an element b ∈ B such that b ∈ A. If this
is true, then A is a proper subset of B. Otherwise, A = B, implying
that A is not a proper subset of B.
1.1.9 The Size of a Set
The size of a set is an important feature of a set.
Definition: If there are exactly n distinct elements in a set S where n
is a nonnegative integer(natural number),we say that S is a finite set
and the size of S is n, which is called the cardinality of S. The cardinality
of S is denoted by | S |.
1.1.10 Power Sets
Idea: the power set of a set S contains every subset of S.
Definition: Given a set S, the power set of S is the set of all subsets
of the set S. The power set of S is denoted by P(S).
Example 1: P(∅) = {∅} (Why?)
Example 2: P({∅}) = {∅, {∅}} (Why?)
1.1.11 Cartesian product
Idea: ordered pair of elements in a set.
Definition: Let A and B be two sets. The Cartesian product of A and
B, denoted by A × B, is the set of all ordered pairs (a, b), where a ∈ A and
b ∈ B. Therefore:
A × B = {(a, b) | a ∈ A and b ∈ B}
The Cartesian product of more than two sets can also be defined similarly:
A1 × A2 × ... × An = {(a1, a2, ..., an) | ai ∈ Ai for i = 1, 2, ..., n}
In specific, if A = A1 = A2 = ... = An, then A1 × A2 × ... × An can also
be written as An
.
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6. Chapter 2
Set Operations
There are several typical set operations.
2.1 the unions of sets
Definition: Let A and B be two sets. The union of the sets A and B, denoted
by A ∪ B, is the set containing those elements that are either in A or in B,
or in both.
In another word, if there is an element x in sets A or B, x will also in the
union of A and B.
A ∪ B = {x | x ∈ A or x ∈ B}
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