Modelling of Computer Systems
Table of Content
Chapter 1 Propositional Logic
1.1 Propositions and Deductions
1.2 The Language of Propositional Logic
1.3 Modelling with Propositional Logic
1.4 Ambiguities of Natural Languages
1.5 Truth Tables
1.6 Equivalences and Valid Arguments
Chapter 2 Sets
2.1 Set Notation
2.2 Membership, Equality and Inclusion
2.3 Set and Properties
2.4 Operations on Sets
2.5 Ordered pairs and Cartesian Products
2.6 Logical Equivalences versus Set Identities
2.7 Example
Chapter 4 Predicate Logic
4.1 Predicates and Free Variables
4.2 Quantifiers and Bound Variables
4.3 Rules of Quantification
4.4 Modelling in Predicate Logic
Chapter 6 Functions
6.1 Definition
6.2 Injective and Surjective
6.3 Composing Functions
Chapter 7 Relation
7.1 Basic Definition
7.2 Binary Relation
7.3 Operations on Binary Relations
7.4 Properties of Binary Relations
1
,Modelling of Computer Systems
Chapter 1 Propositional Logic
1.1 Propositions and Deductions
1. A statement is a declaration which is either true or false
2. Atomic statement is a statement without connectives
1.2 The Language of Propositional Logic
1. Propositional variables P, Q, R represent unknown proposition.
2. Propositional connectives “not” (¬), “or” (∨), and (∧), “implies” (⇒), “equivalent” (⇔) are
used to combine propositions.
Connectives Symbol Pronunciation
Negation ¬p not p、p does not hold、p is false、it is not the case that p......
Disjunction p∨q p or q、p or q or both、p unless q
Conjunction p∧q p and q、p but q、not only p but also q
Implication p⇒q p implies q、p only if q、if p then q、q if p、q whenever p
Equivalence p⇔q p if, and only if, q、p is equivalent to q
3. Propositional formula is represented by either an atomic formula with a variable such
as P, Q, R or a compound formula built up with propositional connectives such as P ∨ Q.
4. Well formed Formula refers to a statement written in propositional logic
5. Parentheses and precedences should be carefully handled.
Connectives are applied right to left, such that p ⇒ q ∨ r ⇒ s is interpreted
as p ⇒ (q ∨ r) ⇒ s due to ∨ binds more tightly than ⇒. It then becomes p ⇒
((q ∨ r) ⇒ s) due to righttoleft order.
6. Syntax Tree is constructed by a wellformed formula.
Note: Consider a statement (P ∨ Q) ∧ R, it is clear that P, Q, R are propositional variables, so
they are propositional formulae. Thus (P ∨ Q) is a propositional formula so (P ∨ Q) ∧ R is
also a propositional formula.
1.3 Modelling with Propositional Logic
1.4 Ambiguities of Natural Languages
2
, Modelling of Computer Systems
1.5 Truth Tables
1.6 Equivalences and Valid Arguments
1. Tautology is a proposition which is true regardless of the truth values
2. Contradiction is a proposition which is false regardless of the truth values
3. Satisfiable is a proposition which is sometimes true or false under some interpretation
3
Table of Content
Chapter 1 Propositional Logic
1.1 Propositions and Deductions
1.2 The Language of Propositional Logic
1.3 Modelling with Propositional Logic
1.4 Ambiguities of Natural Languages
1.5 Truth Tables
1.6 Equivalences and Valid Arguments
Chapter 2 Sets
2.1 Set Notation
2.2 Membership, Equality and Inclusion
2.3 Set and Properties
2.4 Operations on Sets
2.5 Ordered pairs and Cartesian Products
2.6 Logical Equivalences versus Set Identities
2.7 Example
Chapter 4 Predicate Logic
4.1 Predicates and Free Variables
4.2 Quantifiers and Bound Variables
4.3 Rules of Quantification
4.4 Modelling in Predicate Logic
Chapter 6 Functions
6.1 Definition
6.2 Injective and Surjective
6.3 Composing Functions
Chapter 7 Relation
7.1 Basic Definition
7.2 Binary Relation
7.3 Operations on Binary Relations
7.4 Properties of Binary Relations
1
,Modelling of Computer Systems
Chapter 1 Propositional Logic
1.1 Propositions and Deductions
1. A statement is a declaration which is either true or false
2. Atomic statement is a statement without connectives
1.2 The Language of Propositional Logic
1. Propositional variables P, Q, R represent unknown proposition.
2. Propositional connectives “not” (¬), “or” (∨), and (∧), “implies” (⇒), “equivalent” (⇔) are
used to combine propositions.
Connectives Symbol Pronunciation
Negation ¬p not p、p does not hold、p is false、it is not the case that p......
Disjunction p∨q p or q、p or q or both、p unless q
Conjunction p∧q p and q、p but q、not only p but also q
Implication p⇒q p implies q、p only if q、if p then q、q if p、q whenever p
Equivalence p⇔q p if, and only if, q、p is equivalent to q
3. Propositional formula is represented by either an atomic formula with a variable such
as P, Q, R or a compound formula built up with propositional connectives such as P ∨ Q.
4. Well formed Formula refers to a statement written in propositional logic
5. Parentheses and precedences should be carefully handled.
Connectives are applied right to left, such that p ⇒ q ∨ r ⇒ s is interpreted
as p ⇒ (q ∨ r) ⇒ s due to ∨ binds more tightly than ⇒. It then becomes p ⇒
((q ∨ r) ⇒ s) due to righttoleft order.
6. Syntax Tree is constructed by a wellformed formula.
Note: Consider a statement (P ∨ Q) ∧ R, it is clear that P, Q, R are propositional variables, so
they are propositional formulae. Thus (P ∨ Q) is a propositional formula so (P ∨ Q) ∧ R is
also a propositional formula.
1.3 Modelling with Propositional Logic
1.4 Ambiguities of Natural Languages
2
, Modelling of Computer Systems
1.5 Truth Tables
1.6 Equivalences and Valid Arguments
1. Tautology is a proposition which is true regardless of the truth values
2. Contradiction is a proposition which is false regardless of the truth values
3. Satisfiable is a proposition which is sometimes true or false under some interpretation
3
