# Discrete Mathematics Lecture by MikeJenny

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Discrete Structures
Chapter 1 Part B
Fundamentals of Logic
Nurul Amelina Nasharuddin
Multimedia Department

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Predicates
• A predicate is a sentence that contains a
finite number of variables and becomes a
statement when specific values are
substituted for the variables
• The domain of a predicate variable is a set
of all values that may be substituted in place
of the variable
• P(x): x is a student at UPM
• P(x, y): x is a student at y
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Predicates
• The sets in which predicate variables take their values may
be described either in words or in symbols
• x  A indicates that x is an element of the set A
• x  A; x is not an element of the set A
• {1, 2, 3} refers to the set whose elements are 1, 2 and 3
• Certain set of numbers that frequently referred to are
Symbol                         Set                          Example

R      Set of all real numbers                    All points on a line number

Z      Set of all integers                        …,-3, -2, -1, 0, 1, 2, 3, …

Q      Set of all rational numbers, or quotient   {p/q | p ∈ Z, q ∈ Z, q ≠ 0}
of integers
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Predicates
• If P(x) is a predicate and x has domain D, the truth set of
P(x) is the set of all elements in D that make P(x) true
when substituted for x. The truth set is denoted as:
{x  D | P(x)}
which is read “the set of all x in D such that P(x)”
• True or false?: Let P(x): x2 > x with domain set R (real
numbers).

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Predicates
• Let Q(n) be the predicate “n is a factor of
8.” Find the truth set of Q(n) if
– the domain of n is the set of Z+
– the domain of n is the set of Z

• Ans:
•{1, 2, 4, 8}
•{-1, -2, -4, -8, 1, 2, 4, 8}

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Predicates
• Let P(x) and Q(x) be predicates with the common
domain D
• P(x)  Q(x) means that every element in the truth
set of P(x) is in the truth set of Q(x), or
equivalently for every x, P(x)  Q(x)
• P(x)  Q(x) means that P(x) and Q(x) have
identical truth sets, or equivalently for every x,
P(x)  Q(x)

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Universal Quantifier, 
• Let P(x) be a predicate with domain D.
• A universal statement is a statement in the form
“x  D, P(x)”.  denotes “for all”
• It is true iff P(x) is true for every x from D.
It is false iff P(x) is false for at least one x from D.
• A value of x for which P(x) is false is called a
counterexample to the universal statement
• Examples
– D = {1, 2, 3, 4, 5}: x  D, x² >= x        (True)
– x  R, x² >= x                             (False)
• Method of exhaustion                                     7
Existential Quantifier, 
• Let P(x) be a predicate with domain D.
• An existential statement is a statement in the form
“x  D, P(x)”.  denotes “there exists”
• It is true iff P(x) is true for at least one x from D.
It is false iff P(x) is false for every x from D.
• Examples:
– m  Z, m² = m                             (True)
– E = {5, 6, 7, 8, 9}, x  E, m² = m        (False)

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Quantifier
• Rewrite the following without using  and
:
a. x  R, x2  0
b. x  Z such that m2 = m
• Ans:
a. All real numbers have nonnegative squares
The square of any real number is nonnegative
b. m2 =m, for some integer m
Some integers equal their own square
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Universal Conditional Statement
• Universal conditional statement
“x, if P(x) then Q(x)”:
• Eg: x  R, if x > 2, then x2 > 4
• Universal conditional statement is called
vacuously true or true by default iff P(x) is false
for every x in D
• Eg:
When x =1, if 1 > 2, then 12 > 4             (True)
When x = -3, if (-3) > 2, then (-3)2 > 4     (True)
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Negation of Quantified
Statements
• The negation of a universally quantified
statement x  D, P(x) is x  D, ~P(x)
(x  D, P(x))  x  D such that ~P(x)
• The negation of an existentially quantified
statement x  D, P(x) is x  D, ~P(x)
( x  D, P(x))   x  D such that ~P(x)
• Rewrite formally, formal and informal
negation: No politicians are honest
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No politicians are honest
• Formal version:  politicians x, x is not
honest
• Formal negation:  a politician x such that x
is honest
• Informal negation: Some politicians are
honest

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Negation of Quantified
Statements
• The negation of a universal conditional
statement x  D, P(x)  Q(x) is x  D,
P(x)  ~Q(x)
(x, if P(x) then Q(x))  x such that P(x) and ~Q(x)
• Noted before; ~(p  q)  p  ~q
• Write formal negation:  people p, of p is
blond then p has blue eyes
• Ans:  a person such that p is blond and p
does not have blue eyes
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Exercises
• Write negations for each of the following statements:
–   All dinosaurs are extinct
–   No irrational numbers are integers
–   All COBOL programs have at least 20 lines
–   The sum of any two even integers is even
–   The square of any even integer is even

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Multiply Quantified Statements
• Imagine you are in a factory and the guide tell you
“There is person supervising every detail of the
production process”
• Have both existential quantifier There is and
universal quantifier every
• When a statement contains more than one
quantifier, the actions being performed is in the
order in which the quantifier occur.

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Multiply Quantified Statements
• To establish the truth of the statement “x in D,
y in E such P(x, y)”
Pick whatever x from set D , then find element
y in E that “works” for that particular y in P(x,
y)
• To establish the truth of the statement “x in D,
such that y in E, P(x, y)”
Find element x in D that “works” no matter y in
P(x, y)
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Multiply Quantified Statements
• For all positive numbers x, there exists number y
such that y < x
• There exists number x such that for all positive
numbers y, y < x
• For all people x there exists person y such that x
loves y (Informally: Everybody loves somebody)
• There exists person x such that for all people y, x
loves y (Informally: Somebody loves everybody)

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Negation of Multiply Quantified
Statements
• The negation of x, y, P(x, y)
is logically equivalent to x, y, ~P(x, y)
• The negation of x, y, P(x, y)
is logically equivalent to x, y, ~P(x, y)

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Negation of Multiply Quantified
Statements
• Rewrite formally and negate: Everybody
trusts somebody.
• Formally: (a)  people x,  a person y such
that x trusts y.
• Negation: Somebody trusts nobody.
• Formally: (b)  a person x, such that 
people y, x does not trust y.

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Necessary and Sufficient
Conditions, Only If
• x, r(x) is a sufficient condition for s(x)
means: x, if r(x) then s(x)
• x, r(x) is a necessary condition for s(x)
means: x, if s(x) then r(x)
• x, r(x) only if s(x) means: x, if r(x) then
s(x)

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Logical Equivalence and Logical
Implication for Quantified
Statements in One Variable
x  D, (P(x)  Q(x)  (x  D, P(x))  (x  D, Q(x))
x  D, (P(x)  Q(x))  (x  D, P(x))  (x  D, Q(x))
(x  D, P(x))  (x  D, Q(x))  x  D, (P(x)  Q(x))
x  D, (P(x)  Q(x))  (x  D, P(x))  (x  D, Q(x))

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Application: Digital Logic
Circuits
• Digital Logic Circuit is a basic electronic component of a
digital system
• Values of digital signals are 0 or 1 (bits)
• Black Box is specified by the signal input/output table
• Input/output table = truth table
• Three gates: AND-gate, OR-gate, NOT-gate
Application: Digital Logic
Circuits
• Combinational circuit is a combination of
logical gates
• Combinational circuit always correspond to
some Boolean expression, such that
input/output table of a table and a truth table
of the expression are identical

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Application: Digital Logic
Circuits
• Finding Boolean expression for a circuit:
P  ~ (Q  R)

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Application: Digital Logic
Circuits
• Determining output for a given input: Move
from left to right through the diagram,
tracing the action of each gate on the input
signals
• Constructing the input/output table for a
circuit: list the all possible combinations for
input signals, and find the output for each
by tracing through the unit
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Application: Digital Logic
Circuits
• A recognizer is a circuit that outputs 1 for exactly
one particular combination of input signals and
outputs 0’s for all other combinations
• Multiple-input AND and OR gates
• Finding a circuit that corresponds to a given
input/output table:
– Construct equivalent Boolean expression using
disjunctive normal form: for all outputs of 1 construct a
conjunctive form based on the truth table row. All
conjunctive forms are united using disjunction
– Construct a digital logic circuit equivalent to the
Boolean expression
Application: Digital Logic
Circuits
• Given                     • Boolean expression:
(P  Q  R) 
Input       Output

P     Q      R     S

1     1      1     1      (P  Q  R) 
1     1      0     0
(P  Q  R)
1     0      1     1

1     0      0     1      • Circuit:
0     1      1     0

0     1      0     0

0     0      1     0

0     0      0     0
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Application: Digital Logic
Circuits
• Two digital logic circuits are equivalent iff
their input/output tables are identical
• Simplification of circuits by using Theorem
1.1.1 to simplify the Boolean expression
• Another way to simplify circuit by using
– Scheffer stroke, | (NAND, AND-gate followed
by NOT-gate)
– Peirce arrow,  (NOR, OR-gate followed by
NOT-gate)
Exercises
• Construct circuits for (P  Q)  R
• From the given input/output        Input       Output

P    Q     R     S
table below, construct
1    1     1     0

(a) a Boolean expression        1    1     0     1

1    0     1     0
(b) a circuit                   1    0     0     0

0    1    1     1

0    1    0     0

0    0    1     0

0    0    0     0

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