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Discrete Mathematics
Introduction to Logic & Set
Theory
Outline of lecture 1
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Introduction to course
Textbooks
What is formal logic?
Propositional logic
Propositions
Propositional connectives
Formalisation of arguments
EE1J2 - Slide 2
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Propositional Logic
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Set theory
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Basic ideas: propositions, connectives
Formalising statements in ‘natural language’
Formal proofs
Basic ideas: definitions of sets
Relations, functions and equivalence relations
Cardinality, finite, countable and uncountable sets
Predicate Logic
Logic programming
EE1J2 - Slide 3
Tutorial sheets
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One tutorial sheet handed out each week
To be handed in approx 1 week later
Questions and model solutions will appear
on my web pages
EE1J2 - Slide 4
Recommended Books
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John Truss,
“Discrete
Mathematics for
Computer
Scientists”,
Addison-Wesley,
Second Edition,
1999
EE1J2 - Slide 5
Recommended Books (cont.)
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Nimal Nissanke,
“Introductory
Logic and Sets for
Computer
Scientists”,
Addison-Wesley,
1999
EE1J2 - Slide 6
Recommended Books
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New this year:
James A Anderson,
“Discrete
Mathematics with
Combinatorics (2nd
edition)”, PrenticeHall, 2004
EE1J2 - Slide 7
Recommended books (cont.)
Rod Haggarty
“Discrete mathematics for computing”
Addison Wesley
A Chetwynd and P Diggle,
“Discrete Mathematics” ButterworthHeinemann
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EE1J2 - Slide 8
Other books
I have also made use of two other books:
 Geoffrey Finch, “How to study
linguistics”, Macmillan, 1998
 J N Crossley and others, “What is
mathematical logic?”, Oxford University
Press, 1972
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Introduction to logic
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What is logic?
Why is it useful?
Types of logic
 Propositional logic
 Predicate logic
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What is logic?
“Logic is the
beginning of
wisdom, not the
end”
EE1J2 - Slide 11
What is logic?
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Logic n.1. The branch of philosophy
concerned with analysing the patterns of
reasoning by which a conclusion is drawn
from a set of premises, without reference to
meaning or context
(Collins English Dictionary)
EE1J2 - Slide 12
Why study logic?
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Logic is concerned with two key skills,
which any computer engineer or scientist
should have:
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Abstraction
Formalisation
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Why is logic important?
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Logic is a formalisation of reasoning.
Logic is a formal language for deducing
knowledge from a small number of explicitly
stated premises (or hypotheses, axioms, facts)
Logic provides a formal framework for
representing knowledge
Logic differentiates between the structure and
content of an argument
EE1J2 - Slide 14
Logic as formal language
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In this course, logic will be presented as a
formal language
Within that formal language:
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Knowledge can be stated concisely and
precisely
The process of reasoning from that knowledge
can be made rigorous
EE1J2 - Slide 15
What is an argument?
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An argument is just a sequence of
statements.
Some of these statements, the premises, are
assumed to be true and serve as a basis for
accepting another statement of the
argument, called the conclusion
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Deduction and inference
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If the conclusion is justified, based solely on the
premises, the process of reasoning is called
deduction
If the validity of the conclusion is based on
generalisation from the premises, based on strong
but inconclusive evidence, the process is called
inference (sometimes called induction)
This course is concerned only with deduction
EE1J2 - Slide 17
Two examples
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Deductive argument:
“Alexandria is a port or a holiday resort.
Alexandria is not a port. Therefore, Alexandria
is a holiday resort”
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Inductive argument
“Most students who did not do the tutorial
questions will fail the exam. John did not do
the tutorial questions. Therefore John will fail
the exam”
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Some different types of logic
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Historically, a number of types of logic
have been proposed.
In this course we will study
Propositional logic (Boole, 1815-1864)
Predicate logic (Frege 1848-1925)
EE1J2 - Slide 19
Propositional logic
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Simple types of statements, called
propositions, are treated as atomic building
blocks for more complex statements
Alexandria is a port or a holiday resort.
Alexandria is not a port.
Therefore, Alexandria is a holiday resort
EE1J2 - Slide 20
Propositional logic
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Basic propositions in the argument are
p – Alexandria is a port
q – Alexandria is a holiday resort.
In abstract form, the argument becomes
p or q
Not q
Therefore q
EE1J2 - Slide 21
Predicate logic
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Extension of propositional logic
A ‘predicate’ is just a property
Predicates define relationships between
any number of entities using qualifiers:
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 “for all”, “for every”
 “there exists”
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Example
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Let P(x) be the property
‘if x is a triangle then the sum of its internal
angles is 180o”
In predicate logic:
 x P(x)
“For every x such that x is a triangle, the
sum of the internal angles of x is 180o”
EE1J2 - Slide 23
Another example
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Let P(x) be the property
‘x is an integer and x2 = 4’
Then
 x P(x)
“There exists x such that x is an integer and
x2 = 4”
EE1J2 - Slide 24
Newton’s second law of
motion
“for every x”
“of type called object”
“or”
 x: Object  stationary(x)  in-uniform-motion (x)
  f : Force  x is-acted-upon-by f
“there exists an f”
In English: “for every x of a certain type referred to as an
Object, x is stationary, x is in uniform motion, or there is
an f of type Force such that x is acted upon by f”
EE1J2 - Slide 25
 and 
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Remember:
 x ‘for every x’, or ‘for All x’
 x ‘there is an x’ or ‘there Exists an x’
Tip:
Think of  as an upside down ‘A’ (‘for All’)
Think of  as a backwards ‘E’ (‘there Exists’)
EE1J2 - Slide 26
Propositions
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A (atomic, elementary) proposition is the
underlying meaning of a simple declarative
sentence, which is either true or false
The truth or falsehood of a proposition is
indicated by assigning it one of the truth
values T (for true) or F (for false)
EE1J2 - Slide 27
Example propositions
Mammals are warm blooded
The sun orbits the earth
The evergreen forests of Canada consist of
spruce, pine and fir trees
John is taller than Joan
Joan is shorter than John
John is not shorter than Joan
EE1J2 - Slide 28
Sentences which are not
propositions
Over millions of years they build up on top of
one another to form a reef
Can the arctic hare change the colour of its
coat to match its surroundings?
Put down that book!
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Which are propositions?
Can pigs fly?
Pigs can fly
Sparrows can fly
Joe runs faster than Patrick
Patrick runs slower than Joe
Pay your bills on time
The circumference of a circle is equal to four times
its diameter
EE1J2 - Slide 30
Propositional connectives
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These are the words that we use to join
atomic propositions together to form
compound propositions. E.G:
In 1938 Hitler seized Austria, (and) in 1939
he seized former Czechoslovakia and in
1941 he attacked the former USSR while
still having a non-aggression pact with it
EE1J2 - Slide 31
Propositional connectives
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Propositional logic has four connectives
Name
Read as
Symbol
negation
‘not’

conjunction
‘and’

disjunction
‘or’

implication
‘if…then…’

EE1J2 - Slide 32
Interpretation of connectives
Connective
Interpretation
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negation p is true if and only if p is false
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A conjunction pq is true if and only if both p
and q are true
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A disjunction pq is true if and only if p is true
or q is true.
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An implication p  q is false if and only if p is
true and q is false
EE1J2 - Slide 33
Some more terminology…
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Expressions either side of a conjunction are
called conjuncts (pq)
Expressions either side of a disjunction are
called disjuncts (pq)
In the implication p  q, p is called the
antecedent and q is the consequence
EE1J2 - Slide 34
Precedence of connectives
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In complex propositions, brackets may be
used to remove ambiguity.
(p q) r versus p  (q  r)
By convention, the order of precedence
Brackets, Negation, Conjunction,
Disjunction, Implication
EE1J2 - Slide 35
Formalisation
Statement
In 1938 Hitler seized Austria, (and) in 1939 he
seized former Czechoslovakia and in 1941 he
attacked the former USSR while still having a
non-aggression pact with it
EE1J2 - Slide 36
Formalisation (continued)
Atomic propositions:
p – In 1938 Hitler seized Austria
q – In 1939 Hitler seized former Czechoslovakia
r – In 1941 Hitler attacked the former USSR
s – In 1941 Hitler had a non-aggression pact with
the former USSR
Formalisation in Propositional Logic:
pqrs
EE1J2 - Slide 37
Formalisation
Although both Stanley and Gordon are not
young, Stanley has a better chance of winning
the next bowling tournament, despite Gordon’s
considerable experience
EE1J2 - Slide 38
Formalisation (continued)
Atomic propositions:
p – Stanley is young
q – Gordon is young
r – Stanley has a better chance of winning the next bowling
tournament
s – Gordon has considerable experience in bowling
Formalisation in Propositional Logic:
(p)  (q)  r  s
EE1J2 - Slide 39
Negation and atomic
propositions
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Note that for first atomic proposition I
chose:
Stanley is young
and not
Stanley is not young
EE1J2 - Slide 40
Summary of Lecture 1
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Introduction to course
Textbooks
What is formal logic?
Propositional logic
Propositions
Propositional connectives
Formalisation of arguments
EE1J2 - Slide 41