Introduction to Mathematical Logic

Introduction to Mathematical Logic
Sections A, B, C, D
September 23, 2023
1 Propositional Calculus
1.1 Definition
Definition: A proposition is a declarative sentence (or statement) that can be true (T) or false
(F) but not both at the same time.
Notation We use capital letters to denote a proposition ; for example: P,Qor R...
"T" stands for "true".
"F" stands for "False".
T and F are called truth values of P.
Examples
1. "2 is a natural number": is a proposition (simple).
2. "1
2is an irrational number": is proposition.
3. " Where are you?": is not a proposition.
4. "Close the door": is not a proposition.
1.2 Logical connectives (Propositional connectives)
1.2.1 Logical negation or negation
Definition: Let P be a proposition; "not" P , denoted P , is a proposition which has the
following truth table:
P P
T F
F T
1.2.2 Logical Disjunction
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Definition: Disjunction of P and Q , denoted P∨Q , is a proposition which has the
following truth table:
P Q P ∨Q
T T T
T F T
F T T
F F F
P∨Qis read Por Q.
1.2.3 Logical conjunction
Definition: Conjunction of P and Q , denoted P∧Q , is a proposition which has the
following truth table:
P Q P ∧Q
T T T
T F F
F T F
F F F
P∧Qis read Pand Q.
1.2.4 Logical implication
Definition: Let P and Q be propositions; we denote by P=⇒Q a proposition whose
truth table is as follows:
P Q P =⇒Q
T T T
T F F
F T T
F F T
Remark
1. P=⇒Qis read "if Pthen Q".
2. Pis called assumption ;Qis called conclusion.
Definition:
1The proposition Q=⇒Pis called the converse of P=⇒Q.
2The proposition Q=⇒Pis called the contrapositive of de P=⇒Q.
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1.2.5 Logical equivalence
Definition: Let Pand Qpropositions.
We designe by P⇐⇒ Qa proposition whose truth table is as follows:
P Q P ⇐⇒ Q
T T T
T F F
F T F
F F T
P⇐⇒ Qis read Pif and only if Qor Pis equivalent to Q.
1.3 Tautology
Definition: Tautology is a compound proposition which is true for every truth value of the
individual propositions.
Examples Prove that these propositions are tautologies:
1. (P⇐⇒ Q)⇐⇒ ((P=⇒Q)∧(Q=⇒P))
Denote this proposition α; then construct its truth table.
P Q P =⇒Q Q =⇒P(P=⇒Q)∧(Q=⇒P)P⇐⇒ Q α
T T T T T T T
T F F T F F T
F T T F F F T
F F T T T T T
All the truth values of αare true then αis tautology.
2. (P=⇒Q)⇐⇒ (Q=⇒P)
Denote this proposition β; construct its truth table.
P Q P =⇒Q Q =⇒P β
T T T T T
T F F F T
F T T T T
F F T T T
All the truth values of βare true then αis tautology.
Remark Let αand βbe compound propositions;
α⇐⇒ βis a tautology if and only if αand βhave the same truth values.
1.4 Exercice
Prove that these propositions are tautologies:
1. P⇐⇒ P
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2. (P=⇒Q)⇐⇒ (Q=⇒P)
3. (P=⇒Q)⇐⇒ P∨Q.
4. P=⇒Q⇐⇒ P∧Q.
5. P∧Q⇐⇒ P∨Q.De Morgan’s laws
6. P∨Q⇐⇒ P∧Q.De Morgan’s laws
7. (P⇐⇒ Q)⇐⇒ ((P=⇒Q)∧(Q=⇒P))
8. (P∧Q)⇐⇒ (Q∧P),∧is commutative
9. (P∨Q)⇐⇒ (Q∨P),∨is commutative
10. (P∧(Q∨R)) ⇐⇒ ((P∧Q)∨(P∧R)),∧is distributif with respect to ∨
11. (P∨(Q∧R)) ⇐⇒ ((P∨Q)∧(P∨R)),∨is distributif with respect to ∧
1.5 Logical equivalence
If two propositions P and Q have the same truth values , they are called logically equivalent
and we denote this by P≡Q.
Remark
1. Warning : ≡is not a connective.
2. We can see that P≡Qif and only if P⇐⇒ Qis a tautology.
2 Quantified expression
2.1 Introduction to Predicate calculus
Definition: In literature, a predicate is the part of sentence that tells us something about
the subject. It contains the verb.
Exemple
1. The lighthouse was damaged in the storm.
"was damaged in the storm" is the prédicat".
2. Birds are chirping outside the windows .
"are chirping outside the windows" is the predicat.
Definition: In mathematics, a predicate is a statement that contains variables and it may
be true or false depending on the values of theses variables.
The domainof a predicate variable is the set , E , of all values that may be subtituted for the
variables.
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Examples
1. E= The set of real numbers..
P(x):" x > 5".
For x= 1 ,P(1) is false.
For y= 7 ,P(7) is true.
2. E= the set of integers.
P(x, y):" x+yis odd".
P(2,6) is false.
P(3,4) is true.
2.2 Quantifiers
2.2.1 Universal quantifier
The universal quantifier is denoted ∀, it is read "for all" or "for every" or "for each" .
Let Ebe a non empty set, and a predicate P(x)
∀x∈D P (x)
is a proposition which is true if and only if P(x)is true for all values of xin E.
∀x∈D P (x)means every xin Ehas property P.
Example
D=R.
P(x):" x+ 1 > x".
∀x∈RP(x)is a true proposition.
Remark
1. (∀x∈D P (x)) is false if and only if we find a value of xin Ethat P(x)is false.
2. An element x0 in E for which P ( x0 )is false is called a counterexample of ∀x∈
D P (x).
2.2.2 Existential quantifier
Existential quantifier is denoted ∃ , it is read "there exists" or "there is" or "for at least
one" .
Let Ebe a non empty set, and a predicate P(x)
∃x∈D P (x)
is a proposition which is true if and only if there is an xfor which P(x)is true .
Example
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