This document summarizes Chapter 1 of a textbook on basic set theory and functions. It defines what a set is and how to define sets using roster and set builder methods. It introduces important basic sets like natural numbers, integers, rational numbers, and real numbers. It also covers subsets, the size and power set of a set, Cartesian products, and empty sets. Chapter 2 is said to cover typical set operations like union.

1. CS205-Basic Set Theory and Functions
Suiliang Ma
May,2016

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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