LECTURE 7
RMIT University; Taylor's College
T o c o m b i n e
propositions using
connectives
T o c o n s t r u c t t h e
truth table of a given
compound proposition
T o d e f i n e d e M o r g a n
Law for logic
T o d e f i n e t h e
difference between a
predicate and a
proposition
T o u s e a q u a n t i f i e r
in a predicate
1
NEGATION: A REVISION
 Let P be a proposition.
 It has a truth value of T (for true) or F (for false).
 Its negation is “not P”, denoted by ~P.
P
~P
T
F
F
T
When P is true, ~P is false.
When P is false, ~P is true.
Truth table for Negation
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2
COMBINING PROPOSITIONS
Let P and Q be propositions. They can be combined in various
ways.
 Conjunction (AND)
PQ
 Disjunction (OR)
PQ
 Implication (if P then Q)
PQ
 Equivalence (P if and only if Q)
PQ
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3
COMPOUND PROPOSITIONS
 Expressions created using the five connectives
~,,, , 
are called compound propositions.
 We combine elementary (or constituent) propositions
to create compound propositions.
 The truth values of the constituent propositions
determine the truth values of a compound
proposition.
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4
TRUTH TABLES
P
Q
P Q
P
Q
T
T
F
F
T
F
T
F
T
F
F
F
T
T
F
F
T
F
T
F
P
Q
P Q
P
Q
T
T
F
F
T
F
T
F
T
T
F
F
T
F
T
F
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P Q
P Q
5
EXAMPLE 1
a) Construct the truth table for the compound
proposition ~ P  Q
P  Q ~ P  Q
b) Construct the truth table for the compound
proposition ( P  Q)  (Q  P)
P  Q  ( P  Q)  (Q  P)
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6
EXAMPLE 2
a) Construct the truth table for the compound
propositions ( P  Q ) and P  Q.
b) What is the relationship between these two
compound propositions?
( P  Q)  P  Q
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DE MORGAN LAWS FOR LOGIC
( P  Q)  P  Q
( P  Q)  P  Q
 Consequence: Any expression using “or” can be replaced by an
expression using “and” and “not”
 “I’ll watch G. I. Joe or Final Destination 4 this weekend”
is logically equivalent to
 “It’s not true that I won’t watch G.I. Joe and that I won’t watch
Final Destination 4 this weekend”
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8
HIERARCHY OF CONNECTIVES
~,,, , 
 Example 3: Construct the truth table for the following
compound proposition.
PQ  PQ
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TAUTOLOGY, CONTRADICTION,
CONTINGENCY
 If the last column in a truth table has only T (for true), then
the compound proposition is called a tautology
 If the last column in a truth table has only F (for false),
then the compound proposition is called a contradiction
 If a compound proposition is neither a tautology nor a
contradiction then the last column of the truth table will
have both T and F appearing. Such a compound proposition
is called a contingency
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10
EXAMPLE 4
Show that the following is a contradiction.
( P  Q) ~ P
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11
LOGICAL EQUIVALENCES
Several logical equivalences have been established
P  Q ~ P  Q
P  Q  ( P  Q)  (Q  P)
( P  Q)  P  Q
( P  Q)  P  Q
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LOGICAL EQUIVALENCES
 Consequence: All compound propositions can be
expressed using only two connectives: negation and
conjunction, or negation and disjunction, or negation
and implication
 Example: “If dogs have humps then the moon is
green” is equivalent to “Dogs don’t have humps or
the moon is green”.
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13
PROPOSITIONS VS. PREDICATES
 A proposition has to be unambiguously true or false.
 In contrast, a predicate is a statement involving at least one
variable, for example, the variable x. The truth value may
depend on the value of x.
 Example: Let P(x) mean “x is an integer”. Then P(2) is true,
but P(π) is false.
 Example: Let Y(t) mean “my friend t wears glasses”. Then
Y(Albert) may be true, while Y(Aaron) may be false.
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PREDICATE LOGIC
 What values is the variable allowed to take?
 There may be several variables. Example, let T( x, y)
mean x and y are relatively prime.
 The variables have to range over some set D, called
the domain of interpretation or the universe of
discourse.
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EXAMPLE 5
Let the domain of interpretation for a predicate T( x, y)
be D = Z = the set of all integers.
 Let T(x, y) mean that x and y are relatively prime.
 Then T(10, 21) is true while T(12, 15) is false.
 Why?
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QUANTIFIERS
 It’s natural to introduce the idea of a quantifier when we’re
considering predicates. These tell us how often the predicate
is true
 “for all” 
 “there exists”

 Predicate logic involves statements like this:
 xP( x )
[ for all x, P(x) is true ]
 xyP (x )
[ for all x there exists y such that P(x, y) is true ]
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