LOGIC
Mai Phuong Vuong
Lecture 1
Symbolic logics
Mathematical propositions
and truth values
LOGIC
Logical operations
Propositions with quantifiers
Mai Phuong Vuong
School of Applied Mathematics and Informatics
Hanoi University of Science and Technology
1.1
LOGIC
Outline
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.2
1 Symbolic logics
Mathematical propositions and truth values
Logical operations
Propositions with quantifiers
LOGIC
Outline
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.3
1 Symbolic logics
Mathematical propositions and truth values
Logical operations
Propositions with quantifiers
LOGIC
What is a statement?
Mai Phuong Vuong
Definition
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.4
A statement (or a proposition) is a sentence that is true
or false but not both.
1
Statement variables
LOGIC
What is a statement?
Mai Phuong Vuong
Definition
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.4
A statement (or a proposition) is a sentence that is true
or false but not both.
Denoted by letters of alphabet: A, B, . . . , p, q, . . . 1
1
Statement variables
LOGIC
What is a statement?
Mai Phuong Vuong
Example
1
Symbolic logics
2
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
3
4
A: Every natural number is either even or odd.
B: Every natural number is odd.
p: 1 + 1 = 2
π
q: sin
= 0.07
13
The truth value of a statement
• A is true, A = T or A = 1
• B is false, B = F or B = 0
1.5
LOGIC
Logical operations
Mai Phuong Vuong
1
Symbolic logics
Mathematical propositions
and truth values
Logical operations
2
Propositions with quantifiers
3
4
5
1.6
Negation
Conjunction ∧
Disjunction ∨
Conditional →
Bi-conditional ↔
LOGIC
Logical operations
Mai Phuong Vuong
1. Negation
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.7
If A is a statement variable, the negation of A is “not A”
and is denoted A. It is true if A is false, and false if A is
true.
LOGIC
Logical operations
Mai Phuong Vuong
1. Negation
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
If A is a statement variable, the negation of A is “not A”
and is denoted A. It is true if A is false, and false if A is
true.
Example
A
Every natural number is odd
3 is a prime number.
1+1=2
1.7
A
NOT every natural number is odd.
3 is NOT a prime number
1 + 1 ̸= 2
LOGIC
Logical operations
Mai Phuong Vuong
1. Negation: Table of truth values
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.8
A A
1 0
0 1
LOGIC
Logical operations
Mai Phuong Vuong
2. Conjunction
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.9
If A and B are statement variables, the conjunction of A
and B is “A and B,” and is denoted A ∧ B. It is true only
when both A and B are true, otherwise it is false.
LOGIC
Logical operations
Mai Phuong Vuong
2. Conjunction
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
If A and B are statement variables, the conjunction of A
and B is “A and B,” and is denoted A ∧ B. It is true only
when both A and B are true, otherwise it is false.
Example
A
3 is a prime number
1<2
1.9
B
1+1=2
8 is divisible by 2
A∧B
...
...
LOGIC
Logical operations
Mai Phuong Vuong
2. Conjunction: Table of truth values
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.10
A
1
1
0
0
B A∧B
1
1
0
0
1
0
0
0
LOGIC
Logical operations
Mai Phuong Vuong
3. Disjunction
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.11
If A and B are statement variables, the disjunction of A
and B is “A or B,” and is denoted A ∨ B. It is false only
when both A and B are false, otherwise it is true.
LOGIC
Logical operations
Mai Phuong Vuong
3. Disjunction
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
If A and B are statement variables, the disjunction of A
and B is “A or B,” and is denoted A ∨ B. It is false only
when both A and B are false, otherwise it is true.
Example
A
3 is a prime number
π is greater than 4
1.11
B
4 is an odd number
π is a natural number
A∨B
...
LOGIC
Logical operations
Mai Phuong Vuong
3. Disjunction
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.12
A
1
1
0
0
B A∨B
1
1
0
1
1
1
0
0
LOGIC
Logical operations
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.13
Example
Evaluate the truth value of the following compound
statement:
A∧B
LOGIC
Logical operations
Mai Phuong Vuong
4. Conditional
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.14
If A and B are statement variables, the conditional of B
by A is “if A then B,” or “A implies B” and is denoted
A → B. It is false only when A is true and B is false,
otherwise it is true.
LOGIC
Logical operations
Mai Phuong Vuong
4. Conditional
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
If A and B are statement variables, the conditional of B
by A is “if A then B,” or “A implies B” and is denoted
A → B. It is false only when A is true and B is false,
otherwise it is true.
Example
A
3 is a prime number
π is less than 3
1.14
B
4 is an odd number
π is not a natural number
A→B
...
...
LOGIC
Logical operations
Mai Phuong Vuong
4. Conditional
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.15
A
1
1
0
0
B A→B
1
1
0
0
1
1
0
1
LOGIC
Logical operations
Mai Phuong Vuong
5. Biconditional
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.16
If A and B are statement variables, the biconditional of A
and B is “A if and only if B,” and is denoted A ↔ B. It is
true only when A and B have the same truth values, and
is false if A, B have opposite truth values.
LOGIC
Logical operations
Mai Phuong Vuong
5. Biconditional
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
If A and B are statement variables, the biconditional of A
and B is “A if and only if B,” and is denoted A ↔ B. It is
true only when A and B have the same truth values, and
is false if A, B have opposite truth values.
Example
A
3 is a prime number
π is less than 3
1.16
B
4 is an odd number
π is a natural number
A↔B
...
...
LOGIC
Logical operations
Mai Phuong Vuong
5. Biconditional
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.17
A
1
1
0
0
B A↔B
1
1
0
0
1
0
0
1
LOGIC
Logical operations
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.18
A
1
1
0
0
B A ∨B A ∧B A A → B A ↔ B
1
1
0
1
0
0
0
1
1
1
1
0
0
1
1
LOGIC
Statement forms (Logical forms)
Mai Phuong Vuong
Definition
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.19
A statement form is an expression made up ofstatement
variables and logical operations that becomes a
statement when actual statements are substituted for the
component statement variables.
LOGIC
Statement forms (Logical forms)
Mai Phuong Vuong
Definition
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
A statement form is an expression made up ofstatement
variables and logical operations that becomes a
statement when actual statements are substituted for the
component statement variables.
The true table of a statement form
List the truth values correspond to all possible
combinations of truth values for component statement
variables.
1.19
LOGIC
Statement forms (Logical forms)
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
Example
Evaluate the truth value of the following statement form:
(A ∨ B) ∧ A ∧ B
1.20
LOGIC
Statement forms (Logical forms)
Mai Phuong Vuong
Definition (Tautology)
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.21
A tautology (tautological statement) is a statement form
that is always true regardless of the truth values of the
individual statements substituted for its statement
variables.
LOGIC
Statement forms (Logical forms)
Mai Phuong Vuong
Definition (Tautology)
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
A tautology (tautological statement) is a statement form
that is always true regardless of the truth values of the
individual statements substituted for its statement
variables.
Definition (Contradiction)
A contradiction (contradictory statement) is a statement
form that is always false regardless of the truth values of
the individual statements substituted for its statement
variables.
1.21
LOGIC
Statement forms (Logical form)
Mai Phuong Vuong
Example
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.22
Construct the truth tables for the following statement
forms. Which ones are tautologies/contradictions?
1 (A ∧ B) ∨ C
2 (A ∧ B) ∨ (A ∨ (A ∧ B))
3 ((A ∧ B) ∧ (B ∧ C)) ∧ B
LOGIC
Logical equivalence
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
Definition
Two statement forms A and B are called logically
equivalent iff they have identical truth values for each
possible substitution of statement variables.
Denote:
1.23
A⇔B
LOGIC
Logical equivalence
Mai Phuong Vuong
• Identical laws
A∧1⇔A
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.24
A∨0⇔?
LOGIC
Logical equivalence
Mai Phuong Vuong
• Identical laws
A∧1⇔A
Symbolic logics
A∨0⇔?
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
• Universal bound laws
A∨1⇔1
A∧0⇔?
1.24
LOGIC
Logical equivalence
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
• Idempotent laws
A∧A⇔A
A∨A⇔A
A⇔A
1.25
LOGIC
Logical equivalence
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
• Commutative laws
Logical operations
Propositions with quantifiers
1.26
A∧B ⇔B∧A
A∨B ⇔B∨A
LOGIC
Logical equivalence
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
• Associative laws
Logical operations
Propositions with quantifiers
1.27
(A ∧ B) ∧ C ⇔ A ∧ (B ∧ C)
(A ∨ B) ∨ C ⇔ A ∨ (B ∨ C)
LOGIC
Logical equivalence
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
• Distributive laws
Logical operations
Propositions with quantifiers
1.28
A ∧ (B ∨ C) ⇔ (A ∧ B) ∨ (A ∧ C)
A ∨ (B ∧ C) ⇔ (A ∨ B) ∧ (A ∨ C)
LOGIC
Logical equivalence
Mai Phuong Vuong
Symbolic logics
• De Morgan’s laws
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
A∧B ⇔A∨B
A∨B ⇔A∧B
1.29
LOGIC
Logical equivalence
Mai Phuong Vuong
Symbolic logics
• Other laws
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.30
A→B ⇔A∨B
A ↔ B ⇔ (A → B) ∧ (B → A)
LOGIC
Logical equivalence
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.31
Example
Show that the following statement forms are logically
equivalent.
A ↔ B and (A ∧ B) ∨ (A ∧ B)
LOGIC
Quantified statements
Mai Phuong Vuong
Definition (Predicate (or quantified function))
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
A predicate (or quantified function) is a sentence that
contains finite number of variables and becomes a
statement when specific values are substituted for the
variables.
Denote: P(x1 , x2 , . . . , xn ): a predicate of n variables
x1 , x2 , . . . , xn .
1.32
LOGIC
Quantified statements
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Definition (Domain of a predicate variable)
Logical operations
Propositions with quantifiers
1.33
The domain of a predicate variable is the set of all values
that may be substituted in place of the variable.
LOGIC
Quantified statements
Mai Phuong Vuong
Predicates of 1 variable
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.34
3 divides n
LOGIC
Quantified statements
Mai Phuong Vuong
Predicates of 1 variable
Symbolic logics
Mathematical propositions
and truth values
3 divides n
Logical operations
Propositions with quantifiers
⇓
∃n ∈ Z, 3 divides n
∀n ∈ Z, 3 divides n
1.34
LOGIC
Quantified statements
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Definition (Quantifiers)
Logical operations
Propositions with quantifiers
1.35
• Universal quantifier: ∀
• Existential quantifier: ∃
LOGIC
Quantified statements
Mai Phuong Vuong
Predicates of 2 variables
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
x +y =0
⇓
1.36
LOGIC
Quantified statements
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.37
(Quantifierk + Variablek + Domaink ) + Predicate
LOGIC
Quantified statements
Mai Phuong Vuong
Symbolic logics
Example
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.38
Translate from informal language to formal language
1 All real numbers are even.
2 Every real number has a non-negative square.
LOGIC
Quantified statements
Mai Phuong Vuong
Symbolic logics
Example
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.39
Translate from formal language to informal language
2
1 ∀x ∈ R, x ≥ 0
2
2 x ∈ R, x = −1
LOGIC
The negation of a quantified statement
Mai Phuong Vuong
Example
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.40
∀x ∈ X , P(x) ⇔
LOGIC
The negation of a quantified statement
Mai Phuong Vuong
Example
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.40
∀x ∈ X , P(x) ⇔ ∃x ∈ X , P(x)
LOGIC
The negation of a quantified statement
Mai Phuong Vuong
Example
Symbolic logics
Mathematical propositions
and truth values
Logical operations
∀x ∈ X , P(x) ⇔ ∃x ∈ X , P(x)
Propositions with quantifiers
∃x ∈ X , P(x) ⇔
1.40
LOGIC
The negation of a quantified statement
Mai Phuong Vuong
Example
Symbolic logics
Mathematical propositions
and truth values
Logical operations
∀x ∈ X , P(x) ⇔ ∃x ∈ X , P(x)
Propositions with quantifiers
∃x ∈ X , P(x) ⇔ ∀x ∈ X , P(x)
1.40
LOGIC
The negation of a quantified statement
Mai Phuong Vuong
Symbolic logics
Example
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.41
Write the negation of this sentence in informal language
1 All real numbers are even.
2 Every real number has a non-negative square.
LOGIC
The negation of a quantified statement
Mai Phuong Vuong
Symbolic logics
Example
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.42
Translate from formal language to informal language
1 ∀x ∈ R, x < 2.
2 ∀n ∈ Z, n ÷ 2.
LOGIC
The negation of a quantified statement
Mai Phuong Vuong
Symbolic logics
Mathematical propositions
and truth values
Logical operations
Propositions with quantifiers
1.43
Example
Write the negation of the following proposition, then
translate it to informal language.
1 ∀x ∈ R, ∃y ∈, ∃z ∈ R, x + y + z = 0
2 ∃x ∈ R, ∀y ∈ R, x + y = 1 or x + y = −1