Module 1 (Part 1)-Sets and Number Systems.pdf

Math 111-Precalculus
Remegia L. Ganot-Math Faculty
Department of Mathematics &
Statistics, USeP-CAS
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Module 1: Sets and Number
System
Definition & Basic Notation
Subsets & Counting
Operations on Sets
Counting numbers
Integers
Rational and Irrational
numbers
Real numbers and their
properties
Complex Numbers

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Note:
• All throughout the discussions, we use capital letters 𝐴, 𝐵, 𝐶, 𝐷, … , 𝑋, 𝑌, 𝑍
as the names of sets and the lower case 𝑎, 𝑏, 𝑐, 𝑑, … . 𝑥, 𝑦, 𝑧 as the elements
of sets.
• We usually make use of the symbol ℕ for the set of natural numbers but to
avoid confusion to its definition, we use the symbol ℕ here as the set of
nonnegative integers.
Question: Is zero a natural number?
Answer: Please watch the video on youtube.com/watch?v=6upQWEpHhuc
Definition 1.1. A set is a well-defined collection of objects.
LESSON 1: Sets

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METHODS OF DESCRIBING SETS
A. The roster method.
This method describes a set by listing the names of its elements,
separated by commas, with the full list enclosed in braces.
Example 1.2. 𝐴 = 1, 2, 3, 4, 5 ;
𝐵 = 𝐷𝑎𝑣𝑎𝑜, 𝐷𝑖𝑔𝑜𝑠, 𝑃𝑎𝑛𝑎𝑏𝑜, 𝑇𝑎𝑔𝑢𝑚
B. The rule or Description method or set-builder notation.
This method describes a set in terms of one or more properties to be
satisfied by objects in the set, and by those objects only.
Example 1.3. 𝐶 = 𝑥 𝑥 𝑖𝑠 𝑎 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 ≤ 100
𝐷 = 𝑥│𝑥 𝑖𝑠 𝑎 𝑓𝑒𝑚𝑎𝑙𝑒 𝑓𝑎𝑐𝑢𝑙𝑡𝑦 𝑖𝑛 𝑈𝑆𝑒𝑃 − 𝐶𝐴𝑆, 𝑀𝑎𝑡ℎ &𝑆𝑡𝑎𝑡 𝐷𝑒𝑝𝑡.

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Definition 1.4. The set 𝐴 is a 𝑠𝑢𝑏𝑠𝑒𝑡 of the set B, written 𝐴 ⊆ 𝐵, if and only if
every element of 𝐴 is also an element of 𝐵. If, in addition, there is at least one
element of 𝐵 which is not an element of 𝐴, then 𝐴 is a 𝑝𝑟𝑜𝑝𝑒𝑟 𝑠𝑢𝑏𝑠𝑒𝑡 of 𝐵,
and it is written 𝐴 ⊂ 𝐵.
Illustration: 𝑽𝒆𝒏𝒏 𝑫𝒊𝒂𝒈𝒓𝒂𝒎
𝑼 𝑩
𝑨
𝑨 ⊆ 𝑩
LESSON 2: Subsets and Counting

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Example 1.5. Let set 𝐴 = 1,2,3,4,5 and set 𝐵 = 1,2,3,4,5,6,7 . Then set 𝐴 ⊆ 𝐵,
in fact set 𝐴 is a proper subset of set 𝐵, written 𝐴 ⊂ 𝐵.
Definition 1.6. The 𝒆𝒎𝒑𝒕𝒚 𝒔𝒆𝒕 (or 𝒏𝒖𝒍𝒍 𝒔𝒆𝒕) is the set that contains no elements.
The empty set is denoted by the symbol ∅ or { }.
Definition 1.8. Sets 𝑨 and 𝑩 are equal if and only if every element of 𝑨 is also an
element of 𝑩 and every element of 𝑩 is also an element of 𝑨.
Definition 1.9. The concept of "subset" may be used to define what is meant by
two sets being equal, that is, two sets 𝑨 and 𝑩 are said to be equal, written 𝑨=𝑩, if
and only if 𝑨⊆𝑩 and 𝑩⊆𝑨.
Example 1.10. Let set 𝑨={𝒓, 𝒆, 𝒎, 𝒚} and set 𝑩={𝒎, 𝒆, 𝒓,𝒚}. Then sets 𝑨 and 𝑩
are equal.
Example 1.7. The following are empty sets:
(a) 𝑥 ∈ 2ℤ 2 < 𝑥 < 4
(b) 𝑥 ∈ ℤ 𝑥2 + 2 = 0

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Definition 1.11. If 𝐴 is a finite set, we use the symbol 𝑛(𝐴) to represent the
cardinal number of set 𝐴 (or simply the number of elements in 𝐴), where 𝑛 ∅ =
0 and 𝑛 𝐴 = 𝑘 (for 𝐴 ≠ ∅ and 𝑘 a positive integer) if and only if 𝑘 is the
positive integer having the property that the elements in 𝐴 can be matched, in a
one-to-one fashion, with the positive integers 1,2, … , 𝑘.
Example 1.12. In Example 1.10, set 𝑨={𝒓,𝒆,𝒎,𝒚} has 4 elements, hence 𝒏(𝑨)=𝟒.
Definition 1.13. A set with 𝑘 elements has 2𝑘 subsets.
Example 1.14. In Example 1.10, set 𝐴 = 𝑟, 𝑒, 𝑚, 𝑦 has 24
= 16 subsets,
namely:
∅, 𝑟 , 𝑒 , 𝑚 , 𝑦 , 𝑟, 𝑒 , 𝑟, 𝑚 , 𝑟, 𝑦 , 𝑒, 𝑚 , 𝑒, 𝑦 ,
𝑚, 𝑦 , 𝑟, 𝑒, 𝑚 , 𝑟, 𝑚, 𝑦 , 𝑒, 𝑚, 𝑦 , 𝑟, 𝑒, 𝑦 , 𝑟, 𝑒, 𝑚, 𝑦 .
,

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Definition 1.15. Let 𝐴 be a set. We denote by 𝒫(𝐴), the power set of 𝐴, the set of
all subsets of 𝐴.
Example 1.16. The power set of 𝐴 in Example 1.9 is
𝒫 𝐴 = ∅, 𝑟 , 𝑒 , 𝑚 , 𝑦 , 𝑟, 𝑒 , 𝑟, 𝑚 , 𝑟, 𝑦 , 𝑒, 𝑚 ,
It is interesting to note that in Example 1.16, the cardinal number of the
power set of 𝐴 is, 𝑛(𝒫 𝐴 ) = 16.
Let us discuss the structure of 𝒫(𝐴) by answering correctly the following
questions:

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Example 1.17.
(a) Is ∅ ∈ 𝒫(𝐴)?
(b) Is ∅ ∈ 𝒫(𝐴)?
(c) Is ∅ ⊆ 𝒫(𝐴)?
(d) Is 𝑟 ∈ 𝒫(𝐴)?
(e) Is 𝑟 ⊆ 𝒫(𝐴)?
(f) Is 𝑟 ⊆ 𝒫(𝐴)?
(g) Is 𝑚, 𝑒 , 𝑚, 𝑦 ⊆ 𝒫(𝐴)?

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Explanations on the questions of Example 1.17
(a) Is ∅ ∈ 𝓟(𝑨)?
Yes! Because the elements of the power set of 𝐴, denoted by 𝓟(𝑨) contains all the subsets of set 𝐴 and an
empty set denoted by ∅ is a subset of 𝐴. In fact, ∅ is a subset of every set.
(b) Is ∅ ∈ 𝓟(𝑨)?
No! Because ∅ is already a set and being enclosed with another braces has different meaning from the
elements of the power set of 𝐴.
(c) Is ∅ ⊆ 𝓟(𝑨)?
Yes! An empty set denoted by ∅ is an element of 𝓟(𝑨) and when enclosed by braces means that it is a
subset of the power set of 𝐴.
(d) Is 𝒓 ∈ 𝓟(𝑨)?
No! Because the elements of 𝓟(𝑨) are all sets (actually subsets of set 𝐴) but 𝑟 is not a set it is actually an
element of 𝐴.
(e) Is 𝒓 ⊆ 𝓟(𝑨)?
No! Because {𝒓} is an element of 𝓟(𝑨).
(f) Is 𝒓 ⊆ 𝓟(𝑨)?
Yes! {𝒓} is an element of 𝓟(𝑨) and when enclosed by another braces means a subset of 𝓟(𝑨).
(g) Is 𝒎, 𝒆 , 𝒎, 𝒚 ⊆ 𝓟(𝑨)?
Yes! {𝒎,𝒆} and {𝒎,𝒚} are elements of 𝓟(𝑨) and when enclosed by braces means a subset of 𝓟(𝑨).

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Definition 1.18. Set 𝐴 is equivalent to set 𝐵, denoted by 𝐴 ∼ 𝐵, if and only if 𝐴
and 𝐵 have the same cardinal number or number of elements.
Example 1.19. Let set 𝐴 = 1,2,3,4,5 and set 𝐵 = 𝑎, 𝑏, 𝑐, 𝑑, 𝑒 . Then 𝑛 𝐴 = 5
and 𝑛 𝐵 = 5 and so 𝐴 ∼ 𝐵

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Definition 1.20. The union of sets 𝐴 and 𝐵, denoted by 𝐴 ∪ 𝐵, is the set that
contains all the elements that belong to 𝐴 or to 𝐵 or to both, in set-builder
notation, we have
𝐴 ∪ 𝐵 = 𝑥 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵 = 𝑥 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵
Illustration.
𝑼
𝑨 𝑩
𝑨 ∪ 𝑩
LESSON 3: Set Operations

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Illustration.
𝑼
𝑨 𝑩
𝑨 ∪ 𝑩

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Definition 1.21. The intersection of sets 𝐴 and 𝐵, denoted by 𝐴 ∩ 𝐵, is the set of
elements common to both A and B, in set-builder notation, we have 𝐴 ∩ 𝐵 =
𝑥 𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵 = 𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 .
Illustration 𝑼
𝑨 𝑩
𝑨 ∩ 𝑩
Note: Two sets 𝐴 and 𝐵 are disjoint if their intersection is the empty set, that is,
𝐴 ∩ 𝐵 = ∅.

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Definition 1.22. Let 𝐴 be a subset of a universal set 𝑈. We define the complement
of 𝐴, denoted 𝐴𝑐
, by the rule 𝐴𝑐
= 𝑥 ∈ 𝑈 𝑥 ∉ 𝐴 .
Illustration.
𝑼
𝑨
𝑨𝒄 = 𝑼 − 𝑨 = 𝑼 ∩ 𝑨𝒄

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Definition 1.23. Let 𝐴 and 𝐵 be sets. We define the difference
𝐵 − 𝐴 (read "𝐵 minus 𝐴") by the rule 𝐵 − 𝐴 = 𝑥 𝑥 ∈ 𝐵 𝑎𝑛𝑑 𝑥 ∉ 𝐴 .
Illustration.
Note. The difference 𝐵 − 𝐴 is also called the complement of 𝐴 relative to
𝐵 consists of all elements that are elements of 𝐵 and are not elements of A.
𝑼
𝑨 𝑩
𝑩 − 𝑨 = 𝑩 ∩ 𝑨𝒄

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Definition 1.24. Let 𝐴 and 𝐵 be sets. We define the 𝑠𝑦𝑚𝑚𝑒𝑡𝑟ic difference of 𝐴
and 𝐵, de noted 𝐴 ∆ 𝐵, by the rule 𝐴 ∆ 𝐵 = 𝐴 − 𝐵 (𝐵 − 𝐴).
Illustration.
𝑼
𝑨 𝑩
𝑨 ∆ 𝑩

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LESSON 4: Counting Numbers
Definition 1.25. The set of counting numbers is the set {1, 2, 3, ...} which is
commonly called the set of natural numbers; however, other definition include 0,
so that the set of non-negative integers {0, 1, 2, 3, ...} is also called the set of
natural numbers. The set of natural numbers including 0 is also called the set of
whole numbers. They can be positive, negative, or zero.
1.26. Properties of ℕ:
Addition Laws on ℕ = {𝟏, 𝟐, 𝟑, … }. For all 𝑚, 𝑛, 𝑝 ∈ ℕ,
𝐴1. Closure Law: 𝑚 + 𝑛 ∈ ℕ
𝐴2. Commutative Law: 𝑚 + 𝑛 = 𝑛 + 𝑚
𝐴3. Associative Law: 𝑚 + 𝑛 + 𝑝 = 𝑚 + 𝑛 + 𝑝
𝐴4. Cancellation Law: If 𝑚 + 𝑝 = 𝑛 + 𝑝, then 𝑚 = 𝑛.

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Multiplication Laws on ℕ = {𝟏, 𝟐, 𝟑, … }. For all 𝑚, 𝑛, 𝑝 ∈ ℕ,
𝑀1. Closure Law: 𝑚 ∙ 𝑛 ∈ ℕ
𝑀2. Commutative Law: 𝑚 ∙ 𝑛 = 𝑛 ∙ 𝑚
𝑀3. Associative Law: 𝑚 ∙ (𝑛 ∙ 𝑝) = (𝑚 ∙ 𝑛) ∙ 𝑝
𝑀4. Cancellation Law: If 𝑚 ∙ 𝑝 = 𝑛 ∙ 𝑝, then 𝑚 = 𝑛.
Distributive Laws on ℕ = {𝟏, 𝟐, 𝟑, … }. For all 𝑚, 𝑛, 𝑝 ∈ ℕ,
𝐷1. 𝑚 ∙ 𝑛 + 𝑝 = 𝑚 ∙ 𝑛 + 𝑚 ∙ 𝑝
𝐷2. 𝑛 + 𝑝 ∙ 𝑚 = 𝑛 ∙ 𝑚 + 𝑝 ∙ 𝑚
The Order Relations on ℕ = {𝟏, 𝟐, 𝟑, … }. For each 𝑚, 𝑛 ∈ ℕ, we define
" < “ by 𝑚 < 𝑛 if and only if there exists some 𝑝 ∈ ℕ such that
𝑚 + 𝑝 = 𝑛.

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1.27. The Trichotomy Law on ℕ = {𝟏, 𝟐, 𝟑, … }. For each 𝑚, 𝑛 ∈ ℕ, one and
only one of the following is true:
(i) 𝑚 = 𝑛,
(ii) 𝑚 < 𝑛,
(iii) 𝑚 > 𝑛.
Consequences of the order relations on ℕ = {𝟏, 𝟐, 𝟑, … }:
Theorem 1.28. If 𝑚, 𝑛 ∈ ℕ and 𝑚 < 𝑛, the for each 𝑝 ∈ ℕ,
(i) 𝑚 + 𝑝 < 𝑛 + 𝑝,
(ii) 𝑚 ∙ 𝑝 < 𝑛 ∙ 𝑝
Theorem 1.29. If 𝑚, 𝑛 ∈ ℕ and 𝑚 > 𝑛, the for each 𝑝 ∈ ℕ,
(i) 𝑚 + 𝑝 > 𝑛 + 𝑝,
(ii) 𝑚 ∙ 𝑝 > 𝑛 ∙ 𝑝

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Definition 1.30. The relations “less than or equal to” denoted by the symbol “≤”
and “greater than or equal to” denoted by the symbol “≥” are defined as follows:
(a) For 𝑚, 𝑛 ∈ ℕ, 𝑚 ≤ 𝑛 if either 𝑚 < 𝑛 or 𝑚 = 𝑛.
(b) For 𝑚, 𝑛 ∈ ℕ, 𝑚 ≥ 𝑛 if either 𝑚 > 𝑛 or 𝑚 = 𝑛.
Definition 1.31. Let 𝐴 be any subset of ℕ, i.e., 𝐴 ⊆ ℕ. An element 𝑝 of 𝐴 is
called the least element of 𝐴 provided that 𝑝 ≤ 𝑎 for every 𝑎 ∈ 𝐴.
Theorem 1.32. The set ℕ is well ordered.

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LESSON 5: Integers
Definition 1.33. The set ℤ of integers is the set
ℤ = {… , −3, −2, −1, 0, 1, 2, , 3, … }
1.34. Properties of ℤ.
Addition Laws on ℤ = {… , −3, −2, −1, 0, 1, 2, , 3, … } . For all 𝑚, 𝑛, 𝑝 ∈ ℤ,
𝐴1. Closure Law: 𝑚 + 𝑛 ∈ ℤ
𝐴5. There exists an identity element, 0 ∈ ℤ, relative to addition, such
that 𝑛 + 0 = 0 + 𝑛 = 𝑛 for every 𝑛 ∈ ℤ.
𝐴6. For each 𝑛 ∈ ℤ there exists an identity element, −𝑛 ∈ ℤ, such that
𝑛 + −𝑛 = −𝑛 + 𝑛 = 0.

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Multiplication Laws on ℤ = {… , −3, −2, −1, 0, 1, 2, , 3, … }. For all 𝑚, 𝑛, 𝑝 ∈
ℤ,
𝑀1. Closure Law: 𝑚 ∙ 𝑛 ∈ ℤ
𝑀4. Cancellation Law: If 𝑚 ∙ 𝑝 = 𝑛 ∙ 𝑝 and if 𝑝 ≠ 0 ∈ ℤ, then
𝑚 = 𝑛.
𝑀5. There exists an identity element, 1 ∈ ℤ relative to multiplication,
such that 1 ∙ 𝑛 = 𝑛 ∙ 1 = 𝑛 for every 𝑛 ∈ ℤ.
Distributive Laws on ℤ = {… , −3, −2, −1, 0, 1, 2, , 3, … }. For all 𝑚, 𝑛, 𝑝 ∈ ℤ,
𝐷1. 𝑚 ∙ 𝑛 + 𝑝 = 𝑚 ∙ 𝑛 + 𝑚 ∙ 𝑝
𝐷2. 𝑛 + 𝑝 ∙ 𝑚 = 𝑛 ∙ 𝑚 + 𝑝 ∙ 𝑚

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1.35. The Trichotomy Law on ℤ = {… , −3, −2, −1, 0, 1, 2, , 3, … }. For each
𝑎, 𝑏 ∈ ℤ, one and only one of the following is true:
(i) 𝑎 = 𝑏,
(ii) 𝑎 < 𝑏,
(iii) 𝑎 > 𝑏.
When 𝑎, 𝑏, 𝑐 ∈ ℤ, we have
1. 𝑎 + 𝑐 < 𝑏 + 𝑐 if and only if 𝑎 < 𝑏.
2. 𝑎 + 𝑐 > 𝑏 + 𝑐 if and only if 𝑎 > 𝑏.
3. If 𝑐 > 0, then 𝑎 ∙ 𝑐 < 𝑏 ∙ 𝑐 if and only if 𝑎 < 𝑏.
4. If 𝑐 > 0, then 𝑎 ∙ 𝑐 > 𝑏 ∙ 𝑐 if and only if 𝑎 > 𝑏.
5. If 𝑐 < 0, then 𝑎 ∙ 𝑐 < 𝑏 ∙ 𝑐 if and only if 𝑎 > 𝑏.
6. If 𝑐 < 0, then 𝑎 ∙ 𝑐 > 𝑏 ∙ 𝑐 if and only if 𝑎 < 𝑏.

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Theorem 1.36. If 𝑎, 𝑏 ∈ ℤ and if 𝑎 ∙ 𝑏 = 0, then either 𝑎 = 0 or 𝑏 = 0.
Definition 1.37. Subtraction “−” on ℤ is defined by 𝑎 − 𝑏 = 𝑎 + −𝑏 .
Definition 1.38. The absolute value “ 𝑎 ”, of an integer 𝑎 is defined by
𝑎 =
𝑎 if 𝑎 ≥ 0
−𝑎 if 𝑎 < 0 .

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LESSON 6: The Rational Numbers
Definition 1.39. The set ℚ of rational numbers is defined as follows:
ℚ =
𝑎
𝑏
𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0 .
1.40. Properties of ℚ.
Addition Laws on ℚ =
𝑎
𝑏
𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0 . For all 𝑚, 𝑛, 𝑝 ∈ ℚ,
𝐴1. Closure Law: 𝑚 + 𝑛 ∈ ℚ
𝐴5. There exists an identity element, 0 ∈ ℚ, relative to addition, such
that 𝑛 + 0 = 0 + 𝑛 = 𝑛 for every 𝑛 ∈ ℚ.
𝐴6. For each 𝑛 ∈ ℚ there exists an identity element, −𝑛 ∈ ℚ, such that
𝑛 + −𝑛 = −𝑛 + 𝑛 = 0.

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Multiplication Laws on ℚ =
𝑎
𝑏
𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0 . For all 𝑚, 𝑛, 𝑝 ∈ ℚ,
𝑀1. Closure Law: 𝑚 ∙ 𝑛 ∈ ℚ
𝑀4. Cancellation Law: If 𝑚 ∙ 𝑝 = 𝑛 ∙ 𝑝 and if 𝑝 ≠ 0 ∈ ℚ, then
𝑚 = 𝑛.
𝑀5. There exists an identity element, 1 ∈ ℚ relative to multiplication,
such that 1 ∙ 𝑛 = 𝑛 ∙ 1 = 𝑛 for every 𝑛 ∈ ℚ.
Distributive Laws on ℚ =
𝑎
𝑏
𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0 . For all 𝑚, 𝑛, 𝑝 ∈ ℚ,
𝐷1. 𝑚 ∙ 𝑛 + 𝑝 = 𝑚 ∙ 𝑛 + 𝑚 ∙ 𝑝
𝐷2. 𝑛 + 𝑝 ∙ 𝑚 = 𝑛 ∙ 𝑚 + 𝑝 ∙ 𝑚

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1.41. The Trichotomy Law on ℚ =
𝑎
𝑏
𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0 . If 𝑥, 𝑦 ∈ ℚ, one and
only one of the following holds:
(i) 𝑥 = 𝑦,
(ii) 𝑥 < 𝑦,
(iii) 𝑥 > 𝑦.

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LESSON 7 : The Real Numbers
Definition 1.42. The set ℝ of real numbers is defined in interval form as follows:
ℝ = (−∞, +∞).
1.43. Properties of ℝ.
Addition Laws on ℝ = (−∞, +∞). For all 𝑚, 𝑛, 𝑝 ∈ ℝ,
𝐴1. Closure Law: 𝑚 + 𝑛 ∈ ℝ
𝐴5. There exists an identity element, 0 ∈ ℝ, relative to addition, such
that 𝑛 + 0 = 0 + 𝑛 = 𝑛 for every 𝑛 ∈ ℝ.
𝐴6. For each 𝑛 ∈ ℝ there exists an identity element, −𝑛 ∈ ℝ, such that
𝑛 + −𝑛 = −𝑛 + 𝑛 = 0.

30
Multiplication Laws on ℝ = (−∞, +∞). For all 𝑚, 𝑛, 𝑝 ∈ ℝ,
𝑀1. Closure Law: 𝑚 ∙ 𝑛 ∈ ℝ
𝑀4. Cancellation Law: If 𝑚 ∙ 𝑝 = 𝑛 ∙ 𝑝 and if 𝑝 ≠ 0 ∈ ℝ, then
𝑚 = 𝑛.
𝑀5. There exists an identity element, 1 ∈ ℝ relative to multiplication,
such that 1 ∙ 𝑛 = 𝑛 ∙ 1 = 𝑛 for every 𝑛 ∈ ℝ.
Distributive Laws on ℝ = (−∞, +∞). all 𝑚, 𝑛, 𝑝 ∈ ℝ,
𝐷1. 𝑚 ∙ 𝑛 + 𝑝 = 𝑚 ∙ 𝑛 + 𝑚 ∙ 𝑝
𝐷2. 𝑛 + 𝑝 ∙ 𝑚 = 𝑛 ∙ 𝑚 + 𝑝 ∙ 𝑚

31
1.44. The Trichotomy Law on ℝ = (−∞, +∞). If 𝑥, 𝑦 ∈ ℝ, one and only
one of the following holds:
(i) 𝑥 = 𝑦,
(ii) 𝑥 < 𝑦,
(iii) 𝑥 > 𝑦.
Definition 1.45. The absolute value “ 𝑎 ”, of a real number 𝑎 is defined by
𝑎 =
𝑎 if 𝑎 ≥ 0
−𝑎 if 𝑎 < 0 .

32
LESSON 8 : The Complex Numbers
Definition 1.46. The set ℂ of complex numbers is defined as follows:
ℂ = 𝑥 + 𝑖𝑦 𝑥, 𝑦 ∈ ℝ, 𝑖2
= −1 .
1.47. Properties of ℂ.
Addition Laws on ℂ = 𝑥 + 𝑖𝑦 𝑥, 𝑦 ∈ ℝ, 𝑖2 = −1 . For all 𝑚, 𝑛, 𝑝 ∈ ℂ,
𝐴1. Closure Law: 𝑚 + 𝑛 ∈ ℂ
𝐴5. There exists an identity element, 0 ∈ ℂ, relative to addition, such
that 𝑛 + 0 = 0 + 𝑛 = 𝑛 for every 𝑛 ∈ ℂ.
𝐴6. For each 𝑛 ∈ ℂ there exists an identity element, −𝑛 ∈ ℂ, such that
𝑛 + −𝑛 = −𝑛 + 𝑛 = 0.

33
Multiplication Laws on ℂ = 𝑥 + 𝑖𝑦 𝑥, 𝑦 ∈ ℝ, 𝑖2
= −1 . For all 𝑚, 𝑛, 𝑝 ∈ ℂ,
𝑀1. Closure Law: 𝑚 ∙ 𝑛 ∈ ℂ
𝑀4. Cancellation Law: If 𝑚 ∙ 𝑝 = 𝑛 ∙ 𝑝 and if 𝑝 ≠ 0 ∈ ℂ, then
𝑚 = 𝑛.
𝑀5. There exists an identity element, 1 ∈ ℂ relative to multiplication,
such that 1 ∙ 𝑛 = 𝑛 ∙ 1 = 𝑛 for every 𝑛 ∈ ℂ.
Distributive Laws on ℂ = 𝑥 + 𝑖𝑦 𝑥, 𝑦 ∈ ℝ, 𝑖2
= −1 . For all 𝑚, 𝑛, 𝑝 ∈ ℂ,
𝐷1. 𝑚 ∙ 𝑛 + 𝑝 = 𝑚 ∙ 𝑛 + 𝑚 ∙ 𝑝
𝐷2. 𝑛 + 𝑝 ∙ 𝑚 = 𝑛 ∙ 𝑚 + 𝑝 ∙ 𝑚

34
 The set of nonzero complex numbers: ℂ∗
= 𝒛 ∈ ℂ 𝒛 ≠ 𝟎
 If 𝒙 + 𝒊𝒚 = 𝒛 ∈ ℂ, then the modulus of 𝒛 is 𝒓 = 𝒛 = 𝒙𝟐 + 𝒚𝟐.
Recall: In a right triangle as shown in the figure below,
𝒔𝒊𝒏 𝜽 =
𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝒔𝒊𝒅𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒔𝒊𝒅𝒆
𝒄𝒐𝒔 𝜽 =
𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝒔𝒊𝒅𝒆
𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒔𝒊𝒅𝒆
.
Thus, 𝒔𝒊𝒏 𝜽 =
𝒚
𝒓
⟹ 𝒚 = 𝒓 𝒔𝒊𝒏 𝜽 and 𝒄𝒐𝒔 𝜽 =
𝒙
𝒓
⟹ 𝒙 = 𝒓 𝒄𝒐𝒔 𝜽 . Since 𝒛 = 𝒙 + 𝒊𝒚 and by Substitution
Principle, 𝒛 = 𝒓 𝒄𝒐𝒔 𝜽 + 𝒊𝒓 𝒔𝒊𝒏 𝜽 = 𝒓(𝒄𝒐𝒔 𝜽 + 𝒊𝒔𝒊𝒏 𝜽).
 The polar form of 𝒛 ∈ ℂ is 𝒛 = 𝒓 𝒄𝒐𝒔𝜽 + 𝒊𝒔𝒊𝒏𝜽 = 𝒛 𝒄𝒐𝒔𝜽 + 𝒊𝒔𝒊𝒏𝜽 = 𝒛 𝒆𝒊𝜽
where 𝒆𝒊𝜽
= 𝒄𝒐𝒔𝜽 + 𝒊𝒔𝒊𝒏𝜽.

35
Julia Set formed by 𝒇 𝒛 = 𝒛𝟐
+ 𝒄 where 𝒄 = 𝟎. 𝟔𝟕𝟖 + 𝟎. 𝟑𝟏𝟐𝒊
Julia sets are named after Gaston Julia. He was a French mathematician who
discovered Julia sets and first explored their properties. He lived from 1893 to
1978 and his masterpiece on these sets was published in 1918 (Michael
McGoodwin, March 2000).

36
The Mandelbrot Set M
The Mandelbrot set M, discovered by Benoit B. Mandelbrot c. 1979, can be defined as the set of all
values of the parameter c (Mandelbrot uses the character µ) for which the corresponding Julia sets are each
connected, in fact totally connected. Alternatively, it can be defined as the set of values of c for which the
orbits (successive iterations) of z0 = 0+0i remain bounded (Mandelbrot 183, Peitgen et al. 1992 p.
843). The point 0+0i is termed the critical point for Julia sets. This simple test, i.e., the boundedness of
iterations of 0, thus determines whether a Julia set is connected, a result discovered independently by Julia
and by Fatou (Gagliardo).The definition relating to iterations of 0 was the one Mandelbrot actually used in
his initial explorations of M (Mandelbrot 183). M is a connected set, as shown by Douady and Hubbard
(Peitgen et al. 1992 p. 849). However, this statement must apparently be qualified to "at least locally
connected", since there are said to be an infinite number of points in M that are not currently known to be
connected (Michael McGoodwin, March 2000).

38
∅, empty set a set without
elements
𝑎 ∈ 𝐴 ∈ is a lowercase
Greek letter symbol
which means “an
element of ” or
“belongs to”
𝑎 is an element of set
𝐴; 𝑎 belongs to set 𝐴
𝐴 = 1, 2, 3, 4, 5, ;
2 ∈ 𝐴
𝑎 ∉ 𝐴 not an element of;
does not belong to
𝑎 is not an element of
set 𝐴; 𝑎 does not
belong to set 𝐴
𝐴 = 1, 2, 3, 4, 5, ;
6 ∉ 𝐴

39
ℤ set of integers ℤ = … , −2, −1, 0, 1, 2, … −20 ∈ ℤ;
35 ∈ ℤ
ℤ +
set of positive
integers
ℤ +
= 1, 2, 3, 4, … 100 ∈ ℤ +
ℤ −
set of negative
integers
ℤ −
= … , −2, −1 −12 ∈ ℤ −
ℕ set of natural
numbers
ℕ = {0, 1, 2, … } when
dealing with cardinal
number of sets; otherwise
ℕ = {1, 2, … } which is a
counting number
0 ∈ ℕ when
dealing with
cardinal
number of an
empty set

40
|, ∶, ∋, 𝑠. 𝑡. such that so that or such that 𝐴 = 𝑛 𝑛 ∈ ℤ
ℤ∗
set of nonzero
integers
ℤ ∗
= … , −2, −1, 1, 2, …
= 𝑛 ∈ ℤ 𝑛 ≠ 0
±5 ∈ ℤ ∗
ℙ set of prime
numbers
ℙ = 2, 3, 5, 7, 11, 13, …
= {𝑝 ∈ ℕ|𝑝 > 1, 𝑓𝑎𝑐𝑡𝑜𝑟𝑠 𝑜𝑓 𝑝 𝑎𝑟𝑒 1 𝑎𝑛𝑑 𝑝}
17 ∈ ℙ
ℚ set of rational
numbers
ℚ =
𝑎
𝑏
𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0 −
1
2
, 3 ∈ ℚ
ℚ+
set of positive
rational
numbers
ℚ+
= 𝑥 ∈ ℚ 𝑥 > 0 1
3
, 4 ∈ ℚ+

41
ℚ−
set of negative
rational numbers
ℚ−
= 𝑥 ∈ ℚ 𝑥 < 0
−2, −
1
4
∈ ℚ−
ℚ∗
set of nonzero
rational numbers
ℚ∗
= 𝑥 ∈ ℚ 𝑥 ≠ 0
−
1
8
, 5 ∈ ℚ∗
ℚ𝑐
set of irrational
numbers
ℚ𝑐
=
𝑥│𝑥 𝑖𝑠 𝑛𝑜𝑛𝑟𝑒𝑝𝑒𝑎𝑡𝑖𝑛𝑔 𝑎𝑛𝑑
𝑛𝑜𝑛𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔 𝑑𝑒𝑐𝑖𝑚𝑎𝑙
𝜋, 𝑒, 2 ∈ ℚ𝑐
ℝ set of real
numbers
ℝ = 𝑥│𝑥 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑛 𝑡ℎ𝑒
𝑛𝑢𝑚𝑏𝑒𝑟 𝑙𝑖𝑛𝑒 −3,
3
4
, 𝜋, 𝑒,
2 ∈ ℝ

42
ℝ+
set of positive
real numbers
ℝ+
= 𝒙 ∈ ℝ 𝒙 > 𝟎 1
9
, 2, 𝜋, 𝑒, 3 ∈ ℝ+
ℝ− set of negative
real numbers
ℝ−
= 𝑥 ∈ ℝ 𝑥 < 0 −5, − 3, −1, −
1
1000
∈ ℝ−
ℝ∗
set of nonzero
real numbers
ℝ∗
= 𝒙 ∈ ℝ 𝒙 ≠ 𝟎 − 5, −
1
100
,
1
10000
,
2 ∈ ℝ∗
ℂ set of
complex
numbers
ℂ = 𝒙 + 𝒊𝒚 𝒙, 𝒚 ∈ ℝ, 𝒊𝟐
= −𝟏 −2, 0, 1 + 𝑖 ∈ ℂ

43
ℂ∗
set of nonzero
complex
numbers
ℂ∗
= 𝑧 ∈ ℂ 𝑧 ≠ 0 −5, −2 3
− 𝑖, 2, 5 ∈ ℂ∗
𝐴 𝐵 union of sets 𝐴
and 𝐵
𝐴 𝐵 = 𝑥 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ℚ−
ℚ+
= ℚ∗
𝐴⋂𝐵 intersection of
sets 𝐴 and 𝐵
𝐴⋂𝐵 = 𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵 ℝ⋂ℂ = ℝ
𝐴𝑐
complement of
set 𝐴
𝐴𝑐
= 𝑥 ∈ 𝑈 𝑥 ∉ 𝐴 Let 𝑈 = ℝ. Then
ℝ𝑐
= ∅ and ∅𝑐
=
ℝ

44
𝐴 − 𝐵 complement of
𝐵 relative to 𝐴
𝐴 − 𝐵
= 𝑥 𝑥 ∈ 𝐴 ∧ 𝑥 ∉ 𝐵
ℤ − ℤ−
= ℕ
𝐴 △ 𝐵 symmetric
difference of 𝐴
and 𝐵
𝐴 ∆ 𝐵
= 𝐴 − 𝐵 (𝐵 − 𝐴)
Let 𝑈 = ℝ, 𝐴 = 2, +∞
& 𝐵 = (−∞, 7]. Then
𝐴 △ 𝐵 = (−∞, 2]
(7, +∞)
𝐴 ⊆ 𝐵 set 𝐴 is a
subset of set 𝐵
𝐴 ⊆ 𝐵 means that if
𝑥 ∈ 𝐴 ⟹ 𝑥 ∈ 𝐵
ℝ ⊆ ℝ
𝐴 ⊂ 𝐵 set 𝐴 is a
proper subset
of set 𝐵
𝐴 ⊂ 𝐵 means that if
𝑥 ∈ 𝐴 ⟹ 𝑥 ∈ 𝐵
ℝ ⊂ ℂ

45
Example 1.48. Give an examples of finites sets 𝐴, 𝐵, 𝐶, and 𝐷 and a universal set
𝑈 and determine the following:
𝒂 𝑨 ∪ 𝑩
𝒃 𝑨 ∪ 𝑪
𝒄 𝑩 ∪ 𝑪

46
𝒅 𝑨 ∩ 𝑩
𝒆 𝑨 ∩ 𝑪
𝒇 𝑩 ∩ 𝑪
𝒈 𝑨 ∪ (𝑩 ∩ 𝑪)
𝒉 (𝑨 ∪ 𝑩) ∩ (𝑨 ∪ 𝑪)

47
𝒊 𝑨 ∩ (𝑩 ∪ 𝑪)
𝒋 (𝑨 ∩ 𝑩) ∪ (𝑨 ∩ 𝑪)
𝒌 𝑨𝒄
𝒍 (𝑩 ∪ 𝑫)𝒄
𝒎 𝑩𝒄
∩ 𝑫𝒄

48
𝒅 𝑨 − 𝑫
𝒆 𝑼𝒄
𝒇 ∅𝒄
𝒈 𝑨 ∆ 𝑩
𝒉 𝑨 ∆ 𝑪

A B A B A B
A B
C
A
B
A
A A B
C
A B
C
Example 1.49. In each of the Venn diagrams below, describe the shaded area.
A  B A  B 𝑩𝒄
A  B  C
B
B
𝑩 ⊆ 𝑨 & 𝑩 ∩ 𝑨 = 𝑩 𝑨𝒄
or (𝑨 ∪ 𝑩)𝒄
𝐵𝑐
𝐴 ∩ 𝐵𝑐
∩ 𝐶𝑐
𝑜𝑟 𝐴 − (𝐵 ∪ 𝐶) 𝑨 ∩ 𝑩 ∩ 𝑪𝒄
𝐨𝐫 (𝐀 ∩ 𝑩) − 𝑪
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
49

50
Note: In Slide No. 64,
 𝐴 − 𝐵 ∪ 𝐶 = 𝐴 ∩ 𝐵 ∪ 𝐶 𝑐
= 𝐴 ∩ 𝐵𝑐
∩ 𝐶𝑐
because by De Morgan’s Law
𝐵 ∪ 𝐶 𝑐
= 𝐵𝑐
∩ 𝐶𝑐
.
 𝐴 ∩ 𝐵 − 𝐶 = 𝐴 ∩ 𝐵 ∩ 𝐶𝑐.
For all finite set 𝑨 and 𝑩, and a finite universal set 𝑼:
 𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵 − 𝑛 𝐴 ∩ 𝐵
 𝑛 𝐴 + 𝑛 𝐴𝑐 = 𝑛 𝑈
Example 1.50. In the Venn diagram in the next slide, 𝑈 is the number of people
that attended a local council meeting, A = {people that voted}, and B = {people
that asked for tea}.

51
(a) How many people asked for tea?
(b) How many people asked for tea and voted?
(c) How many people neither asked for tea nor voted?
(d) How many people attended the meeting?
𝑼
𝐴 = {people that
voted}
𝐵 = {people that
asked for tea}
𝟑𝟔 𝟑𝟎 𝟐𝟑
𝟐𝟎
𝒏 𝑩 = 𝟐𝟑 + 𝟑𝟎 = 𝟓𝟑
𝒏 𝑨 ∩ 𝑩 = 𝟑𝟎
𝒏(𝑨 ∪ 𝑩)𝒄= 𝟐𝟎
𝒏 𝑼 = 𝟑𝟔 + 𝟑𝟎 + 𝟐𝟑 + 𝟐𝟎 = 𝟏𝟎𝟗

52
Example 1.51. A music teacher has surveyed 495 students. The results of the
survey are listed below:
320 students like rap music;
395 students like rock music;
295 students like heavy metal music;
280 students like both rap music and rock music;
190 students like both rap music and heavy metal music;
245 students like both rock music and heavy metal music;
160 students like all three.
How many students:
1. like exactly two of the three types of music?
2. like only rock music?
3. like only one of the three types of music?

53
Venn Diagram
𝑼 𝑨 = {𝒍𝒊𝒌𝒆𝒔 𝑹𝒂𝒑 𝑴𝒖𝒔𝒊𝒄} 𝑩 = {𝒍𝒊𝒌𝒆𝒔 𝑹𝒐𝒄𝒌 𝑴𝒖𝒔𝒊𝒄}
𝑪 = {𝒍𝒊𝒌𝒆𝒔 𝑯𝒆𝒂𝒗𝒚 𝑴𝒆𝒕𝒂𝒍 𝑴𝒖𝒔𝒊𝒄}
10 120 30
160
85
30
20
40

54
How many students
1. like exactly two of the three types of music?
Ans: 𝑛 𝐴 ∩ 𝐵 + 𝑛 𝐴 ∩ 𝐶 + 𝑛 𝐵 ∩ 𝐶 = 120 + 30 + 85 = 235
Thus, there are 235 students like exactly two of the three types of music.
2. like only rock music?
Ans: There are 30 students like only rock music.
3. like only one of the three types of music?
Ans: 10 + 30 + 20 = 60.
Thus, there 60 students like only one of the three types of music.

55
Example 1.52. A survey of 1250 Internet users shows the following results concerning the use of
the search engines Google, Bing, Yahoo!, and Ask.
585 use Google.
620 use Yahoo!.
560 use Ask.
450 use Bing.
100 use only Google, Yahoo!, and Ask.
41 use only Google, Yahoo!, and Bing.
50 use only Google, Ask, and Bing.
80 use only Yahoo!, Ask, and Bing.
55 use only Google and Yahoo!.
34 use only Google and Ask.
45 use only Google and Bing.
50 use only Yahoo! and Ask.
30 use only Yahoo! and Bing.
45 use only Ask and Bing.
60 use all four.

56
Use the Venn diagram to determine how many of the Internet users
1. use only Google?
2. use exactly three of the four search engines?
3. do not use any of the four search engines?
𝑼
Google
Yahoo!
Ask
Bing
𝟓𝟓
𝟏𝟎𝟎
𝟔𝟎
𝟒𝟏
𝟐𝟎𝟒
5𝟎
8𝟎
𝟑𝟎
𝟐𝟎𝟎
𝟑𝟒
5𝟎
𝟒𝟓
𝟏𝟒𝟏
𝟒𝟓
𝟗𝟗
𝟏𝟔
Ans: 585 − 55 + 100 + 34 + 60 + 50 + 41 + 45 = 200
Ans: 80 + 41 + 50 + 100 = 271
Ans: 1250 − 141 + 45 + 99 + 204 + 50 + 80 + 30 + 55 + 100 + 60 + 41 + 200 + 34 + 50 + 45 = 16

57
Definition 1.53. The Cartesian product of two sets 𝐴 and 𝐵 can be defined as the
set of all ordered pairs (𝑎, 𝑏) such that 𝑎 𝜖 𝐴 𝑎𝑛𝑑 𝑏 ∈ 𝐵 and is denoted by 𝐴 × 𝐵,
and it can be written as
𝐴 × 𝐵 = (𝑎, 𝑏) 𝑎 ∈ 𝐴 𝑎𝑛𝑑 𝑏 ∈ 𝐵 .
Example 1.54. Let 𝐴 = 𝑎, 𝑏, 𝑐 and 𝐵 = 1, 2 .
Illustration
𝒂
𝒃
𝒄
𝟏
𝟐
𝟏
𝟐
𝟏
𝟐
Then 𝑨 × 𝑩 = 𝒂, 𝟏 , 𝒂, 𝟐 , 𝒃, 𝟏 , 𝒃, 𝟐 , 𝒄, 𝟏 , (𝒄, 𝟐) .
Note : 𝑛 𝐴 × 𝐵 = 𝑛 𝐴 𝑛 𝐵 = 3 2 = 6.

58
Theorems of Set Theory
FACT 1. The following basic laws of set equality or of subsets can be proved to be
theorems of set theory. For all sets 𝐴, 𝐵, and 𝐶 and any universal set 𝑈:
1. 𝐴 = 𝐴 (Reflexive Property of Equality)
2. 𝐴 ⊆ 𝐴 (Reflexive Property of Subset Relation)
3. If 𝐴 = 𝐵 then 𝐵 = 𝐴. (Symmetric Property of Equality)
4. 𝐴 = 𝐵 if and only if 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴. (includes Antisymmetric Property of
Subset)
5. If 𝐴 = 𝐵 and 𝐵 = 𝐶, then 𝐴 = 𝐶 (Transitive Property of Equality)
6. If 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐶, then 𝐴 ⊆ 𝐶 (Transitive Property of Subset Relation)
7. ∅ ⊆ 𝐴
8. 𝐴 ⊆ 𝑈

59
FACT 2. The following basic properties for union and intersection can be
proved to be theorems of set theory. For all sets 𝑨, 𝑩, and 𝑪 and any
universal set 𝑼:
9. 𝐴 ∪ 𝐴 = 𝐴 (Idempotent Law for Union)
10. 𝐴 ∩ 𝐴 = 𝐴 (Idempotent Law for Intersection)
11. 𝐴 ∪ ∅ = 𝐴 (Identity for Union)
12. 𝐴 ∩ 𝑈 = 𝐴 (Identity for Intersection)
13. 𝐴 ∩ ∅ = ∅
14. 𝐴 ∪ 𝑈 = 𝑈
15. 𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴 (Commutative Law for Union)
16. 𝐴 ∩ 𝐵 = 𝐵 ∩ 𝐴 (Commutative Law for Intersection)

60
17. 𝐴 ∪ (𝐵 ∪ 𝐶) = (𝐴 ∪ 𝐵) ∪ 𝐶 (Associative Law for Union)
18. 𝐴 ∩ (𝐵 ∩ 𝐶) = (𝐴 ∩ 𝐵) ∩ 𝐶 (Associative Law for
Intersection)
19. 𝐴 ⊆ 𝐴 ∪ 𝐵
20. 𝐴 ∩ 𝐵 ⊆ 𝐴
FACT 3. The following basic properties for set complement can be proved to
be theorems of set theory. For all sets 𝑨, 𝑩, and 𝑪 and any universal set 𝑼:
21. (𝐴𝑐)𝑐= 𝐴 (Law of Double complementation)
22. 𝐴 ∪ 𝐴𝑐
= 𝑈
23. 𝐴 ∩ 𝐴𝑐
= ∅

61
24. 𝑈𝑐
= ∅
25. ∅𝑐
= 𝑈
FACT 4. The following Distributive Laws can be proved to be theorems of set
theory. For all sets 𝐴, 𝐵, and 𝐶 and any universal set 𝑈:
26. 𝐴 ∪ 𝐵 ∩ 𝐶 = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶) (Union over Intersection)
27. 𝐴 ∩ 𝐵 ∪ 𝐶 = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶) (Intersection over Union)
28. 𝐴 ∩ 𝐵 △ 𝐶 = (𝐴 ∩ 𝐵) △ (𝐴 ∩ 𝐶) (Intersection over Symmetric Difference)
FACT 5. The following basic properties for set difference can be proved to be
theorems of set theory. For all sets 𝐴, 𝐵, and 𝐶 and any universal set 𝑈:
29. 𝐴 − 𝐵 = 𝐴 ∩ 𝐵𝑐
30. 𝐴 − ∅ = 𝐴

62
31. ∅ − 𝐵 = ∅
32. 𝐴𝑐
− 𝐵𝑐
= 𝐵 − 𝐴
33. 𝐴 − 𝐵 − 𝐶 = 𝐴 − 𝐶 − (𝐵 − 𝐶)
FACT 6. The following De Morgan’s Laws can be proved to be theorems of set
theory. For all sets 𝐴, 𝐵, and 𝐶 and any universal set 𝑈:
34. (𝐴 ∩ 𝐵)𝑐
= 𝐴𝑐
∪ 𝐵𝑐
35. (𝐴 ∪ 𝐵)𝑐
= 𝐴𝑐
∩ 𝐵𝑐
36. 𝐴 − 𝐵 ∪ 𝐶 = (𝐴 − 𝐵) ∩ (𝐴 − 𝐶)
37. 𝐴 − 𝐵 ∩ 𝐶 = (𝐴 − 𝐵) ∪ (𝐴 − 𝐶)

63
FACT 7. The following miscellaneous statements of equality or a subset
relationship can be proved to be theorems of set theory. For all sets 𝐴, 𝐵, and 𝐶 and
any universal set 𝑈:
38. 𝐴 = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐵𝑐
)
39. 𝐴 = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐵𝑐)
40. 𝑈 = 𝐴 ∩ 𝐵 ∪ 𝐴𝑐 ∩ 𝐵 ∪ 𝐴 ∩ 𝐵𝑐 ∪ 𝐴𝑐 ∩ 𝐵𝑐
41. 𝐴 ∪ 𝐵 − 𝐴 = 𝐴 ∪ 𝐵
42. (𝐴 − 𝐵)𝑐
= 𝐴𝑐
∪ 𝐵
43. 𝐴 △ 𝐵 = 𝐵 △ 𝐴 (Commutative of Symmetric Difference)
44. 𝐴 △ 𝐵 △ 𝐶 = (𝐴 △ 𝐵) △ 𝐶 (Associativity of Symmetric Difference)
45. 𝐴 △ 𝐴 = ∅
46. 𝐴 △ 𝑈 = 𝐴𝑐

64
47. 𝐴 △ ∅ = 𝐴
48. 𝐴 △ 𝐵 = 𝐴 − 𝐵 ∪ 𝐵 − 𝐴 = 𝐴 ∪ 𝐵 − (𝐴 ∩ 𝐵)
49. 𝐵 × ∅ = ∅ × 𝐶 = ∅
50. 𝐴 ∪ 𝐵 × 𝐶 = (𝐴 × 𝐶) ∪ (𝐵 × 𝐶)
51. 𝐴 ∩ 𝐵 × 𝐶 = (𝐴 × 𝐶) ∩ (𝐵 × 𝐶)
52. 𝐴 − 𝐵 × 𝐶 = 𝐴 × 𝐶 − (𝐵 × 𝐶)
FACT 8. The following statements of equivalence, that is, involving “if and only
if”, can be proved to be theorems of set theory. For all sets 𝐴, 𝐵, and 𝐶 and any
universal set 𝑈:
53. 𝐴 ⊆ 𝐵 if and only if 𝐵𝑐 ⊆ 𝐴𝑐.

65
54. 𝐴 ⊆ 𝐵 if and only if 𝐴 ∪ 𝐵 = 𝐵.
55. 𝐴 ⊆ 𝐵 if and only if 𝐴 ∩ 𝐵 = 𝐴.
56. 𝐴 ⊆ 𝐵 if and only if 𝐴 − 𝐵 = ∅.
57. 𝐴 ⊆ 𝐵 if and only if 𝐴 ∩ 𝐵𝑐 = ∅.
58. 𝐴 ⊆ 𝐵 if and only if 𝐴𝑐
∪ 𝐵 = 𝑈.
FACT 9. The following statements of implication, that is, involving “if…then”,
can be proved to be theorems of set theory. For all sets 𝐴, 𝐵, and 𝐶 and any
universal set 𝑈:
59. If 𝐴 ⊆ 𝐵 and 𝐴 ⊆ 𝐶, then 𝐴 ⊆ 𝐵 ∩ 𝐶.
60. If 𝐴 ⊆ 𝐶 and 𝐵 ⊆ 𝐶, then 𝐴 ∪ 𝐵 ⊆ 𝐶.
61. If 𝐴 ⊆ 𝐵, then 𝐵 = 𝐴 ∪ (𝐵 − 𝐴).

66
62. If 𝐴 ⊆ 𝐶, then 𝐴 ∪ 𝐵 ∩ 𝐶 = (𝐴 ∪ 𝐵) ∩ 𝐶.
63. If 𝐴 ∩ 𝐵 = 𝐴 ∩ 𝐶 and 𝐴 ∪ 𝐵 = 𝐴 ∪ 𝐶, then 𝐵 = 𝐶.
64. If 𝐴 ∩ 𝐵 = 𝐴 ∩ 𝐶 and 𝐴𝑐
∩ 𝐵 = 𝐴𝑐
∩ 𝐶, then 𝐵 = 𝐶.
65. If 𝐴 ∪ 𝐵 = 𝐴 ∪ 𝐶 and 𝐴𝑐 ∪ 𝐵 = 𝐴𝑐 ∪ 𝐶, then 𝐵 = 𝐶.
66. If 𝐴 ∩ 𝐵 = ∅, then 𝐴 △ 𝐵 = 𝐴 ∪ 𝐵.
67. If 𝐴 × 𝐵 = 𝐴 × 𝐶 and 𝐴 ≠ ∅, then 𝐵 = 𝐶.
68. If 𝐴 × 𝐵 = 𝐵 × 𝐴, 𝐴 ≠ ∅ and 𝐵 ≠ ∅, then 𝐴 = 𝐵 .
69. If 𝐴 × 𝐶 = ∅, then 𝐵 = ∅ or 𝐶 = ∅.

67
Definition 1.55. A set 𝐼, all of whose elements are real numbers, is called an
𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 if and only if, whenever 𝑎 and 𝑏 are elements of 𝐼 and 𝑐 is a real
number with 𝑎 < 𝑐 < 𝑏, then 𝑐 ∈ 𝐼.
Definition 1.56. Nine types of intervals are described by the following
terminology and notation, in which 𝑎 and 𝑏 denote real numbers are defined as
follows:
(a) 𝑥 ∈ ℝ 𝑎 ≤ 𝑥 ≤ 𝑏 , a closed and bounded interval, denoted 𝑎, 𝑏 ;
(b) 𝑥 ∈ ℝ 𝑎 < 𝑥 < 𝑏 , an open and bounded interval, denoted 𝑎, 𝑏 ;

68
(c) 𝑥 ∈ ℝ 𝑎 ≤ 𝑥 < 𝑏 , a closed-open and bounded interval, denoted [𝑎, 𝑏);
(d) 𝑥 ∈ ℝ 𝑎 < 𝑥 ≤ 𝑏 , an open-closed and bounded interval, denoted (a, b];
(e) 𝑥 ∈ ℝ 𝑥 ≥ 𝑎 , a closed and unbounded above interval, denoted [𝑎, ∞);

69
(f) 𝑥 ∈ ℝ 𝑥 ≥ 𝑎 , a closed and unbounded above interval, denoted (𝑎, ∞);
(g) 𝑥 ∈ ℝ 𝑥 ≤ 𝑏 , a closed and unbounded below interval, denoted (−∞, 𝑏];
(h) 𝑥 ∈ ℝ 𝑥 < 𝑏 , an open and unbounded below interval, denoted (−∞, 𝑏);

70
(i) ℝ itself is an interval, denoted ℝ1 = −∞, ∞ .
MODULE 1 - EXERCISES
1. Insert ∈ or ∉ in the blank to make the statement correct.
(a) 121 ____ ℚ𝑐
(c)
25
7
____ ℚ (e) 2021 _____ ℕ
(b) −3 ____ ℂ (d) −16 ____ ℝ (f) 0 _____ ℤ∗

71
2. Insert either ⊆ or ⊈ to make the statement correct.
(a) ℕ ____ ℚ (c) 2, 𝜋, 0.333 _____ ℚ𝑐 (e) ℂ ____ ℝ
(b) ℚ ____ ℤ (d) − 9, 0,1 _____ ℚ (f) − 5, 0, 2 ____ ℚ𝑐
3. Determine which of the sets ℕ, ℤ, ℚ, ℚ𝑐
, ℝ, and ∅ is equal to the given set.
(a) ℂ ∩ ℝ = _______ (e) ℚ𝑐 ∪ ℚ = _______
(b) ℤ ∪ ℚ = _______ (f) ℚ ∪ ℂ = ________
(c) ℕ ∩ ℙ = _______ (g) ℚ𝑐
∩ ℚ = _______
(d) ℚ𝑐
∩ ℂ = ______ (h) ∅ ∩ ℕ = _________

72
4. Show the set on the real number line and represent the set by interval notation.
(a) 𝑥 𝑥 > 10 𝑎𝑛𝑑 𝑥 ≤ 20
(b) 𝑥 𝑥 ≤ −1 𝑜𝑟 𝑥 > 3
(c) 𝑥 𝑥 ≥ −6 ∩ 𝑥 𝑥 ≤ 25
(d) 𝑥 𝑥 ≤ 2 ∪ 𝑥 𝑥 ≥ 5

73
5. Show the interval on the real number line and use set-builder notation and
inequality symbols to denote the interval.
(a) [−12,7)
(b) (−∞, 3]
(c) (−1, ∞)

74
6. Use set notation and one or more of the symbols <, >, ≤, and ≥, denote the set,
simplify and show the graph on the real number line:
(a) The set of all 𝑥 such that 𝑥 is between −2 and 2.
(b) The set of all 𝑡 such that 4𝑡 − 1 is nonnegative.

75
(c) The set of all 𝑦 such that 𝑦 + 3 is positive and less than or equal to 15.
(d) The set of all 𝑧 such that 2𝑧 is greater than or equal to -5 and less than −1.

76
7. Show the set on the real-number line and represent the set by interval notation.
𝑎 𝑥 −8 < 𝑥 ≤ 4
𝑏 𝑥 𝑥 ≥ 3 𝑎𝑛𝑑 𝑥 < 15
𝑐 𝑥 𝑥 ≤ −7 𝑜𝑟 𝑥 ≥ 4

77
𝑑 𝑥 𝑥 > −9 ∩ 𝑥 𝑥 ≤ 21
𝑒 𝑥 𝑥 ≤ 0 ∪ 𝑥 𝑥 > 3

78
8. Show the interval on the real-number line and use set notation and inequality
symbols to denote the interval:
𝑎 [−3,7)
𝑏 [−1,5]
𝑐 (0, 6]

79
𝑑 [−5,7)
𝑒 (1, ∞)
𝑓 (−∞, 3]

80
PROBLEM SET 1
In each of the Theorems of Set Theory, do the
following:
1. Give an example using finite sets.
2. Give an example using real number sets,
that is, interval sets (except when the
theorem deals with cross product).

81
REFERENCES
[1] Barnett, Raymond A., Ziegler, Michael R., Byleen, Karl E., Sobecki, D.
Precalculus 7th Edition. McGraw-Hill, c 2011
[2] Hart, William L. Plane and Spherical Trigonometry. Boston: D.C. Heath and
Company, c1964
[3] Johnson, Richard E., et. al. Algebra and Trigonometry 2nd edition. California:
Addison – Wesley Publishing Company, c1971
[4] Leithold, Louis College Algebra and Trigonometry. Massachusetts: Addison –
Wesley Publishing Company, c1989
[5] Miller, Charles D. Fundamentals of College Algebra. New York: Harper
Collins College Publishers, c1994
[6] Robinson N. Elements of Plane and Spherical Trigonometry. American Book
Company, c1970
[7]Spiegel, Murray, Moyer Robert E. College Algebra. New York. McGraw –
Hill, c1998

82
[8] Sullivan, Michael. Trigonometry: A Unit Circle Approach. Prentice Hall, c
2012
[9] Vance, Elbridge P. Modern Algebra and Trigonometry. Massachusetts:
Addison – Wesley Publishing Company, c1975

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