CHAPTER 1 : BASIC LOGIC AND PROOFS
CHAPTER 1
BASIC LOGIC AND
PROOFS
Summary of the Chapter
1.1 Derive Propositional Logic
1.1.1 Define the purpose of proposition logic
1.1.2 Carry out the formulae in proposition logic:
a. Negation
b. Conjunction
c. Disjunction
d. Conditional
e. Biconditional
f. Tautology
1.1.3 Identify the compound proposition
1.1.4 Construct the truth table
1.1.5 Write a well-formed proposition logic in English
1.2 Derive Predicate Logic
1.2.1 Define predicates
1.2.2 State the expression of predicate in a statement
1.2.3 Identify the compound statement in predicate logic
1.2.4 Compare the type of quantifier in: Universal; Existential
1.2.5 Identify the quantified statements
1.2.6 Write a well-formed predicate logic in English
1.2.7 Transfer the translation with quantifiers
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CHAPTER 1 : BASIC LOGIC AND PROOFS
1.3 Demonstrate Proofs
1.3.1 Define theorem and proofs
1.3.2 Identify the logical equivalence rules
1.3.3 Use the rules of inference to validate arguments:
a. Modus Ponens
b. Modus Tollens
c. Hypothetical Syllogism
d. Disjunctive Syllogism
e. Addition
1.3.4 Utilize the rules of inference
1.3.5 Show the proofs using rules of inference
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Conceptual Map
BASIC LOGIC AND PROOF
Derive
Propositional Logic
Derive Predicate
Logic
Demonstrate
Proofs
Purpose of
proposition logic
Define predicates
Theorem and
proofs
Formulae in
proposition logic
Expression of
predicate in a
statement
Logical
equivalence rules
Compound
proposition
Compound
statement in
predicate logic
Use the rules of
inference
Truth table
Type of quantifier
Utilize the rules of
inference
Well-formed
proposition logic
Quantified
statements
Show the proofs
using rules of
inference
Well-formed
predicate logic
Transfer the
translation with
quantifiers
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CHAPTER 1 : BASIC LOGIC AND PROOFS
1.1 Derive Propositional Logic
1.1.1 Define the purpose of proposition logic
The rules of logic are used to distinguish between valid and invalid mathematical
arguments. Besides the importance of logic in understanding mathematical
reasoning, logic has numerous applications to computer science. These rules are
used in the design of computer circuits, the construction of computer programs, the
verification of the correctness of programs, and in many other ways. Furthermore,
software systems have been developed for constructing some, but not all, types
of proofs automatically.
PROPOSITION is a declarative sentence that is either true or false, but not both.
Example 1-1
All the following declarative sentences are propositions:
1) Georgetown is the capital city of Penang.
2) Cats and dogs are fruits.
3) Two plus one equal to three.
4) 1st May is the Labour Day.
Example 1-2
All the following declarative sentences are NOT propositions:
1)
2)
3)
4)
La la la la…
What time is it?
Read this carefully.
x+1=2
We use letters to denote proposition. The letters used for proposition such as p,
q, r, s...
The truth value of a proposition is true, denoted by T, if it is a true proposition,
and the truth value of a proposition is false, denoted by F, if it is a false
proposition.
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Practice 1-1
Which of these are propositions? What is the truth value?
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
The moon is made of green cheese.
4+x=5
There is no pollution in Penang.
2+1=3
What is your name?
Sit down!
Malaysia have snow.
Cats do not fly.
Tiger and snake are herbivores.
2+7=8
Do not trespassing that area.
Red and blue are the colours of the Sun.
Hello!
Malaysia got snow on March.
4-y
Everybody stays calm.
Human cannot live in Jupiter.
1.1.2 Carry out the formulae in proposition logic
Propositional logic studies the ways statements can interact with each other. A
propositional consists of propositional variables and connectives. The connectives
connect the propositional variables.
In propositional logic generally we use five connectives which are −
• OR (∨)
• AND (∧)
• Negation/ NOT (¬ or ∼)
• Implication / if-then (→)
• If and only if (⇔).
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CHAPTER 1 : BASIC LOGIC AND PROOFS
1.1.3 Identify the compound proposition
Many statements are constructed by combining one or more propositions. New
propositions, called compound propositions, are formed from existing propositions
using logical operators.
1. Negation
Let p be a proposition. The negation of p, denoted by ¬ p, is the statement “It is
not the case that p.” The proposition ¬p is read “not p.” The truth value of the
negation of p, is the opposite of the truth value of p. Table 1 displays the truth
table for the negation of a proposition p.
p
T
F
¬p
F
T
Table1
Example 1-3
p = I have a brown hair.
The negation can be written as
¬p = It is not the case that I have a brown hair.
Or more simply as
¬p = I do not have a brown hair.
2. Conjunction
Let p and q be propositions. The conjunction of p and q, denoted by p ∧ q, is the
proposition “p and q.” In logic, the word “but” sometimes is used instead of “and”
in a conjunction. Table 2 displays the truth table of p ∧ q.
p
T
T
F
F
q
T
F
T
F
pq
T
F
F
F
Table 2
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Example 1-4
p = I will have salad for lunch.
q = I will have steak for dinner.
The conjunction of these propositions, p ∧ q, is
p ∧ q = I will have salad for lunch and I will have steak for dinner.
Example 1-5
p = Flash is fast.
q = Superman is faster.
The conjunction of these propositions, p ∧ q, is
p ∧ q = Flash is fast but Superman is faster.
3. Disjunction
Let p and q be propositions. The disjunction of p and q, denoted by p ∨ q, is the
proposition “p or q.” Table 3 displays the truth table for p ∨ q.
p
T
T
F
F
q
T
F
T
F
pq
T
T
T
F
Table 3
Example 1-6
p = I will have salad for lunch.
q = I will have steak for dinner.
The disjunction of these propositions, p ∨ q, is
p ∨ q = I will have salad for lunch or I will have steak for dinner.
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CHAPTER 1 : BASIC LOGIC AND PROOFS
4. Exclusive or
Let p and q be propositions. The exclusive or of p and q, denoted by p ⊕ q.
Table 4 displays the truth table for p ⊕ q.
p
T
T
F
F
q
T
F
T
F
pq
F
T
T
F
Table 4
Example 1-7
p = Students who have taken calculus can enroll in this class.
q = Students who have taken algorithms can enroll in this class.
The exclusive or of these propositions, p ⊕ q, is
p ⨁ q = Students who have taken calculus or algorithms, but not both, can
enroll in this class.
There are several ways to express exclusive or. Another way is
p ⨁ q = Students who have taken either calculus or algorithms, but not
both, can enroll in this class.
5. Conditional
Let p and q be propositions. The conditional statement p → q is the proposition
“if p, then q.”
p
T
T
F
F
q
T
F
T
F
p→q
T
F
T
T
Table 5
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Example 1-8
p = You study hard.
q = You will get a good grade.
p→q = If you study hard, then you will get a good grade.
There are several ways to express conditional. Another way is
p→q = You study hard implies you will get a good grade.
6. Biconditional
Let p and q be propositions. The biconditional statement 𝑝 ↔ 𝑞 is the
proposition “p if and only if q.”
p
T
T
F
F
q
T
F
T
F
𝑝↔𝑞
T
F
F
T
Table 6
Example 1-9
p = You can take the flight.
q = You buy a ticket.
p ↔ q = You can take the flight if and only if you buy a ticket.
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CHAPTER 1 : BASIC LOGIC AND PROOFS
1.1.4 Construct the truth table
We have introduced six important logical operators/connectives. We can use
these connectives to build up complicated compound propositions involving any
number of propositional variables. We can use truth tables to determine the truth
values of these compound propositions. We use a separate column to find the
truth value of each compound expression that occurs in the compound proposition
as it is built up. The truth values of the compound proposition for each combination
of truth values of the propositional variables in it is found in the final column of
the table. Number of rows in the table depends on the number of propositional
variables.
Number of rows = 2𝑛 , where n is the number of propositional variables
Example 1-10
Construct the truth table of the compound proposition (p ˅ ¬ q)→(p ˄ q)
Step 1: Count how many variables are there. The compound proposition has two
variables p and q. By using formula, there are 2 variables, therefore, there must
be four rows in the truth table
Step 2: Assign value in the column.
The 1st and 2nd columns are used for the truth values of p and q.
1st column 2nd column
𝑝
𝑞
T
T
T
F
F
T
F
F
On 3rd column, we put the truth value of ¬q.
1st column 2nd column 3rd column
𝑝
𝑞
¬𝑞
T
T
F
T
F
T
F
T
F
F
F
T
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CHAPTER 1 : BASIC LOGIC AND PROOFS
On the 4th column we assign p ˅ ¬q.
1st column 2nd column 3rd column
𝑝
𝑞
¬𝑞
T
T
F
T
F
T
F
T
F
F
F
T
4th column
𝑝 ˅ ¬𝑞
T
T
F
T
On 5th column we put p ∧ q.
1st column 2nd column 3rd column
𝑝
𝑞
¬𝑞
T
T
F
T
F
T
F
T
F
F
F
T
4th column
𝑝 ˅ ¬𝑞
T
T
F
T
5th column
𝑝˄𝑞
T
F
F
F
Finally, we put (p ˅ ¬q)→(p ˄ q) on the 6th column.
1st column 2nd column 3rd column 4th column 5th column
𝑝
𝑞
¬𝑞
𝑝 ˅ ¬𝑞
𝑝˄𝑞
T
T
F
F
T
F
T
F
F
T
F
T
T
T
F
T
T
F
F
F
6th column
ሺ𝑝 ˅ ¬𝑞ሻ
→ ሺ𝑝 ˄ 𝑞 ሻ
T
F
T
F
Example 1-11
Construct the truth table for ሺ𝑞 → ¬𝑝ሻ ↔ 𝑝
𝑝
𝑞
¬𝑝
𝑞 → ¬𝑝
ሺ𝑞 → ¬𝑝ሻ ↔ 𝑝
T
T
F
F
F
T
F
F
T
T
F
T
T
T
F
F
F
T
T
F
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Example 1-12
Construct the truth table of the compound proposition (p ˅ q)→r
𝑝
𝑞
𝑟
ሺ𝑝 ˅ 𝑞 ሻ
ሺ𝑝 ˅ 𝑞 ሻ → 𝑟
T
T
T
T
T
T
T
F
T
F
T
F
T
T
T
T
F
F
T
F
F
T
T
T
T
F
T
F
T
F
F
F
T
F
T
F
F
F
F
T
Practice 1-2
1. Construct a truth table for each of these compound proposition
a) ሺ𝑝 ˅ 𝑞ሻ → ሺ𝑝 ˄ 𝑞ሻ
b) ሺ𝑝 ˅ ¬𝑞ሻ ↔ 𝑞
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CHAPTER 1 : BASIC LOGIC AND PROOFS
c) ሺ¬𝑝 ⨁ 𝑞ሻ ∧ ¬𝑟
d) ሺ𝑝 ˅ ¬𝑞ሻ ↔ ሺ𝑞 ˄ 𝑟ሻ
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Tautology, Contradiction and Contingency
Mathematicians normally use a two-valued logic: Every statement is
either True or False. A statement in sentential logic is built from simple statements
using the logical connectives such as , , ,
, and
. The truth or falsity
of a statement built with these connectives depends on the truth or falsity of its
components.
A truth table shows how the truth or falsity of a compound statement depends on
the truth or falsity of the simple statements from which it's constructed. Some
propositions are interesting since their values in the truth table are always the
same.
TAUTOLOGY - A compound proposition that is always TRUE, no matter what the
truth values of the propositional variables that occur in it
Example
𝒑
¬𝒑
𝒑 ˅ ¬𝒑
T
F
T
F
T
T
CONTRADICTION - A compound proposition that is always false
Example
𝒑
¬𝒑
𝒑 ˄ ¬𝒑
T
F
F
F
T
F
CONTINGENCY - A compound proposition that is neither a tautology nor a
contradiction
Example
𝒑
¬𝒑
𝒑 → ¬𝒑
T
F
F
F
T
T
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Practice 1-3
Determine whether the following is tautology, contradiction or contingency.
a) ሺ𝑝 ˄ 𝑞ሻ → 𝑝
b) ¬ሺ𝑝 → 𝑞ሻ → ¬𝑞
c) ሺ𝑝 ˄ 𝑞ሻ → ሺ¬𝑝 ˄ 𝑞ሻ
d) ሺ¬𝑝 ˅ 𝑞ሻ ˅ ሺ𝑝 ˄ ¬𝑞ሻ
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CHAPTER 1 : BASIC LOGIC AND PROOFS
1.1.5 Write a well-formed proposition logic in English
Translating from English to Propositional Logic, or from Propositional Logic to
English, takes practice. To do it efficiently is an art. Thankfully, you can do it
systematically. Once you learn the semantics of propositional logic, translation
will become easier. If the sentence you are translating is a simple declarative
sentence, then you’re done.
1. Identify the simple, declarative sentences in it.
2. Translate the identified simple, declarative sentences into atomic
propositions. You can write each one like this: “p = …” or “Let p = …”
3. Identify those words that string the simple sentences together in English,
e.g.: and, but, or, if…then, if and only if, just in case, it is not the case that,
unless, only if, when, etc.
4. Translate the words into the appropriate connectives in propositional logic.
5. Determine the order the atomic propositions combine in with the
connectives.
6. Remember to use the parentheses or precedence of logical operators.
Precedence of Logical Operators
Precedence of operators helps us to decide which operator will get evaluated
first in a complicated looking compound proposition.
Operators
Names
Precedence
()
Bracket
1
¬
Negation
2
^
Conjunction
3
∨
Disjunction
4
→
Implication
5
↔
Biconditional
6
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Practice 1-4
1. Let p and q be the propositions “I write a book” and “I drink latte” respectively.
Express each of these compound propositions as an English sentence.
a. ¬p
b. p ˄ q
c. ¬q → p
d. p → ¬q
e. p ↔ q
f. p ˅ ¬q
2. Let p, q and r be the propositions
p : She writes a letter.
q : I travel to Jordan.
r : The flight ticket is cheap.
Write these propositions by using variables p, q, r and logical operators
a. I travel to Jordan but the flight ticket is not cheap.
b. If she does not write a letter, then I travel to Jordan.
c. I will either travel to Jordan, or she writes a letter, but not both.
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CHAPTER 1 : BASIC LOGIC AND PROOFS
d. The flight ticket is cheap if and only if she writes a letter.
e. The flight ticket is not cheap, or she writes a letter implies I am not travel to
Jordan.
f.
If the flight ticket is cheap then I travel to Jordan.
g. She writes a letter and the flight ticket is cheap.
3. Given the statement as below.
p : I feel healthy.
q : I should stay at home.
r : I have fever.
s : I drink plenty of fluid.
t : I eat nutritious food.
Express each of these propositions as an English sentence.
a. ¬t
b. (¬p ˄ r) → q
c. r → s ˄ t
d. r ˄ ¬q
e. p ↔ (s ˅ t)
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CHAPTER 1 : BASIC LOGIC AND PROOFS
4. Given the statement as below.
p : It is raining.
q : Anne is sick.
r : Kuala Lumpur is the capital of Malaysia.
s : Peter is a loud-mouth.
t : Alan stayed up late last night.
Write these propositions by using variables p, q, r, s, t and logical operators
a. Anne isn’t sick.
b. Alan stayed up late last night and Peter is a loud-mouth.
c. Kuala Lumpur isn’t the capital of Malaysia and it isn’t raining.
d. Kuala Lumpur is the capital of Malaysia and it is raining or Peter is a loudmouth.
e. Anne is sick and it is raining implies that Alan stayed up late last night.
f. It is raining if and only if Anne is sick.
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CHAPTER 1 : BASIC LOGIC AND PROOFS
1.2 Derive Predicate Logic
1.2.1 Define predicates
Propositional logic cannot adequately express the meaning of all statements in
mathematics and in natural language.
For example, “Every person who is 18 years or older, is eligible to vote”
The above statement cannot be expressed using only propositional logic. The
problem is that propositional logic is not expressive enough to deal with
quantified variables. It would have been easier if the statement were referring
to a specific person. But since the statement applies to all people who are 18
years or older, we are stuck.
Therefore, in this section we will introduce a more powerful type of logic called
predicate logic. To understand predicate logic, we first need to introduce the
concept of a predicate and the notion of quantifiers.
A predicate is a sentence that contains a finite number of variables and becomes
a statement when specific values are substituted for the variables.
The domain of a predicate variable is the set of all values that may be substituted
in place of the variable.
1.2.2 State the expression of predicate in a statement
The statement “x is greater than 3” has two parts.
✓ The first part - the variable “x”, is the subject of the statement.
✓ The second part - the predicate, “is greater than 3”, refers to a
property that the subject of the statement can have.
We can denote the statement “x is greater than 3” by P(x), where P denotes the
predicate “is greater than 3” and x is the variable.
The statement P(x) is also said to be the value of the propositional function P at
x. Once a value has been assigned to the variable x, the statement P(x) becomes
a proposition and has a truth value.
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1.2.3 Identify the compound statement in predicate logic
Let say we have “x is greater than 3” and “x is an even number”. If we combine
both statements, we can have a compound statement, such as “If all numbers are
greater than 3, then all numbers are an even number”. How can we write the
compound statement into predicate logic?
1.2.4 Compare the type of quantifier in: Universal; Existential
Quantifiers are words that refer to quantities such as some or all and tell for how
many elements a given predicate is true.
In English, the words all, some, many, none, and few are used in quantifications.
We will focus on two types of quantification here:
1. Universal quantifier, the symbol ∀, denotes “for all”, “for every”, “for each”
Another way to express the sentence
“All human beings are mortal”
is to write
“∀ human beings x, x is mortal.”
If you let H be the set of all human beings, then you can symbolize the statement
more formally by writing
∀x ∈ H, x is mortal
which is read as “For all x in the set of all human beings, x is mortal”
2. Existential quantifier, the symbol ∃, denotes “there exists”, “for some”, “there
is a”
For example, the sentence
“There is a student in class”
can be written as
“∃ a person p such that p is a student in class”
or, more formally,
“∃p ∈ P" such that p is a student in class
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1.2.5 Identify the quantified statements
If we found the words such as all, some, many, none, few, etc. in a statement, so
that statement is a quantified statement.
Equivalences Involving Quantifiers
Two logical statements involving predicates and quantifiers are considered
equivalent if and only if they have the same truth value. There are two very
important equivalences involving quantifiers, given below
1.
∀x (P(x) ∧ Q(x)) ≡ ∀x P(x) ∧ ∀x Q(x)
2.
∃x (P(x) ∨ Q(x)) ≡ ∃x P(x) ∨ ∃x Q(x)
However,
1.
∀x (P(x) ∨ Q(x)) ≢ ∀x P(x) ∨ ∀x Q(x)
2.
∃x (P(x) ∧ Q(x)) ≢ ∃x P(x) ∧ ∃x Q(x)
1.2.6 Write a well-formed predicate logic in English
Translating from logical expressions to English sentences is easy and direct as we
know the meaning of the quantifier’s symbol and other logical operators in the
expression. Let us discuss some examples. While translating, we should identify
the following items:
1.
2.
3.
4.
Types of Quantifiers: Universal (∀) or Existential (∃)
Subject: Variables x, y, etc
Predicate: P(x), Q(x), M(x), etc
Logical Operators: ¬, ∨, ∧, etc
Example 1-13
Let P(x) be the statement “x drops the subject” where the domain for x consists of
all students. Express each of these quantifications in English
a) ∃𝑥 𝑃ሺ𝑥ሻ
There is a student who has drop the subject
b) ∀𝑥 𝑃ሺ𝑥ሻ
Every student drops the subject
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Example 1-14
Let P(x) be the statement “x is an officer” where the domain for x consists of all
people.
Express each of these quantifications in English
a) ∃𝑥 ¬𝑃ሺ𝑥ሻ
There is a people who is not an officer.
b) ∀𝑥 ¬𝑃ሺ𝑥ሻ
Every people is not an officer.
c) ¬∃𝑥 𝑃ሺ𝑥ሻ
It is not the case that there is people who is an officer.
d) ¬∀𝑥 𝑃ሺ𝑥ሻ
It is not the case that all people is an officer.
Example 1-15
Translate these statements into English, where 𝑅ሺ𝑥ሻ is “𝑥 𝑖𝑠 𝑎 𝑟𝑎𝑏𝑏𝑖𝑡” and 𝐻ሺ𝑥ሻ
is “𝑥 ℎ𝑜𝑝𝑠” and the domain consists of all animals.
a) ∀𝑥 (𝑅ሺ𝑥ሻ → 𝐻 ሺ𝑥ሻ)
If all animals is a rabbit then all animals hops 
b) ∀𝑥 (𝑅ሺ𝑥ሻ ∧ 𝐻 ሺ𝑥ሻ)
For all animals such that it is a rabbit and it hops
All animals are a rabbit and all animals hops 
c) ∀𝑥 (𝑅ሺ𝑥ሻ ∨ 𝐻 ሺ𝑥ሻ)
All animals are a rabbit or hops
All animals are a rabbit or all animals hops 
d) ∃𝑥 (𝑅ሺ𝑥ሻ → 𝐻 ሺ𝑥ሻ)
If some animals are a rabbit then some animals hops
e) ∃𝑥 (𝑅ሺ𝑥ሻ ∨ 𝐻 ሺ𝑥ሻ)
There exists an animal such that it is a rabbit or it hops
Some animals are a rabbit or some animals hops 
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CHAPTER 1 : BASIC LOGIC AND PROOFS
f) ∃𝑥 (𝑅ሺ𝑥ሻ ∧ 𝐻 ሺ𝑥ሻ)
There exists an animal such that it is a rabbit and it hops
Some animals is a rabbit and some animals hops 
Example 1-16
Let 𝐿ሺ𝑥, 𝑦ሻ be the statement “𝑥 𝑙𝑜𝑣𝑒𝑠 𝑦” where the domain for x and y consist
of all people.
Express each of these quantifications in English
a) ∀𝑥 ∃𝑦 𝐿ሺ𝑥, 𝑦ሻ
Everybody loves somebody.
b) ∃𝑥 ∀𝑦 𝐿ሺ𝑥, 𝑦ሻ
Somebody loves everybody.
c) ∀𝑥 𝐿ሺ𝑥, 𝑆𝑎𝑚𝑎𝑛𝑡ℎ𝑎ሻ
Everybody loves Samantha.
d) ∃𝑦 𝐿ሺ𝑂𝑙𝑖𝑣𝑒𝑟, 𝑦ሻ
Oliver loves somebody.
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CHAPTER 1 : BASIC LOGIC AND PROOFS
1.2.7 Transfer the translation with quantifiers
Translating sentences in English into logical expressions is a crucial task in
mathematics, logic programming, artificial intelligence, software engineering, and
many other disciplines.
Translating from English to logical expressions becomes even more complex when
quantifiers are needed. We will use some examples to illustrate how to translate
sentences from English into logical expressions.
Example 1-17
Express the statement “Every student in this class has studied calculus” using
predicates and quantifiers
Solution:
✓ First, we rewrite the statement so that we can clearly identify the
appropriate quantifiers to use.
“For every student in this class, that student has studied calculus.”
✓ Next, we introduce a variable x so that our statement becomes
“For every student x in this class, x has studied calculus.”
✓ Then, we introduce C(x), which is the statement “x has studied calculus.”
✓ Therefore, if the domain for x consists of the students in the class, we can
translate our statement as ∀x C(x).
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Example 1-18
Express the statements “For every person x, if person x is a student in this class
then x has studied calculus” using predicates and quantifiers.
Solution:
If S(x) represents the statement that “person x is in this class”, and C(x) be “x has
studied calculus” we see that our statement can be expressed as ∀x(S(x) → C(x)).
Example 1-19
Express the statement “Every student in this class has visited Canada or Mexico”
using predicates and quantifiers.
Solution:
We let C(x) be “x has visited Canada” and M(x) be “x has visited Mexico.” We
see that if the domain for x consists of the students in this class, this statement can
be expressed as ∀x(C(x) ∨ M(x)).
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CHAPTER 1 : BASIC LOGIC AND PROOFS
Practice 1-5
1. Let 𝑃ሺ𝑥ሻ be the statement “x can speak Japanese” and 𝑄ሺ𝑥ሻ be the statement
“x knows cooking.” Express each of these sentences in terms of
𝑃ሺ𝑥ሻ, 𝑄ሺ𝑥ሻ, quantifiers, and logical connectives. The domain for quantifiers
consists of all students.
a) All students can speak Japanese.
b) There is a student who can speak Japanese and who knows cooking.
c) There is a student who can speak Japanese but doesn’t know cooking.
d) If every student can speak Japanese then every students knows cooking.
e) No student knows cooking.
f) All students cannot cook.
g) No students can speak Japanese or knows cooking.
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2. Let C(x) be the statement “x is an accountant” and let D(x) be the statement
“x owns a Porsche” Express each of these statements in terms of C(x), D(x),
quantifiers, and logical connectives. Let the domain consist of all people.
a) Some people own a Porsche.
b) Everyone is an accountant and owns a Porsche.
c) There are people who is an accountant but do not own a Porsche.
d) Nobody is an accountant
3. Let 𝑃ሺ𝑥ሻ be the statement “𝑥 spends five hours in class,” where the domain
for 𝑥 consists of all students. Express each of these quantifications in English.
a) ∃𝑥 𝑃ሺ𝑥ሻ
b) ∀𝑥 𝑃ሺ𝑥ሻ
c) ∃𝑥 ¬𝑃ሺ𝑥ሻ
d) ∀𝑥 ¬𝑃ሺ𝑥ሻ
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4. Translate these statements into English, where 𝐶ሺ𝑥ሻ is “𝑥 is Batman” and
𝐹ሺ𝑥ሻ is “𝑥 is rich” and the domain consists of all people
a) ∀𝑥 ሺ𝐶 ሺ𝑥ሻ → 𝐹ሺ𝑥ሻሻ
b) ∀𝑥 ሺ𝐶 ሺ𝑥ሻ ˄ 𝐹ሺ𝑥ሻሻ
c) ∃𝑥 ሺ𝐶 ሺ𝑥ሻ → 𝐹ሺ𝑥ሻሻ
5. Let 𝑄ሺ𝑥, 𝑦ሻ be the statement “𝑥 sent flowers to 𝑦,” where the domain for
both 𝑥 and 𝑦 consists of all students. Express each of these quantifications in
English.
a) ∃𝑥 ∃𝑦 𝑄ሺ𝑥, 𝑦ሻ
b) ∃𝑥 ∀𝑦 𝑄ሺ𝑥, 𝑦ሻ
c) ∃𝑥 𝑄ሺ𝑥, 𝐿𝑖𝑙𝑦ሻ
d) ∀𝑦 𝑄ሺ𝑅𝑜𝑛𝑎𝑙𝑑, 𝑦ሻ
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