Chapter 1: (Part 2):
The Foundations: Logic and Proofs

Propositional Equivalence
(Section 1.2)

Predicates & Quantifiers
(Section 1.3)
© by Kenneth H. Rosen, Discrete Mathematics & its Applications, Sixth Edition, Mc Graw-Hill, 2007
1
Propositional Equivalences (1.2)

A tautology is a proposition which is always true .
Classic Example: P V P

A contradiction is a proposition which is always
false .
Classic Example: P  P

A contingency is a proposition which neither a
tautology nor a contradiction.
Example: (P V Q)  R
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
2
Propositional Equivalences (1.2) (cont.)

Two propositions P and Q are logically equivalent if
P  Q is a tautology. We write:
PQ
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Propositional Equivalences (1.2) (cont.)

Example:
(P  Q)  (Q  P)  (P  Q)

Proof:


The left side and the right side must have the same truth
values independent of the truth value of the component
propositions.
To show a proposition is not a tautology: use an
abbreviated truth table


try to find a counter example or to disprove the assertion.
search for a case where the proposition is false
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Propositional Equivalences (1.2) (cont.)

Case 1: Try left side false, right side true
Left side false: only one of PQ or Q P need be
false.
1a. Assume PQ = F.
Then P = T , Q = F. But then right side PQ = F.
Wrong guess.
1b. Try Q P = F. Then Q = T, P = F. Then
PQ = F. Another wrong guess.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Propositional Equivalences (1.2)

Case 2. Try left side true, right side false
If right side is false, P and Q cannot have the same truth
value.
2a. Assume P =T, Q = F.
Then PQ = F and the conjunction must be false so the left
side cannot be true in this case. Another wrong guess.
2b. Assume Q = T, P = F.
Again the left side cannot be true. We have exhausted all
possibilities and not found a counterexample. The two
propositions must be logically equivalent.
Note: Because of this equivalence, if and only if or iff is
also stated as is a necessary and sufficient condition for.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Equivalence
Name
PTP
PVFP
Identity Laws
PVTT
PFF
Domination Laws
PVPP
PPP
Idempotent Laws
 ( P)  P
Double Negation
Law
PVQQVP
PQQP
Commutative Law
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
Equivalence
Name
(P V Q) V R
 P V (Q V R)
Associative Law
P V (Q  R)
 (P V Q)  (P V R)
Distributive Law
(P  Q)  P V Q
(P V Q)  P  Q
De Morgan’s Laws
P  Q  P V Q
Implication
Equivalence
P  Q  Q  P
Contrapositive Law
7
Note: equivalent expressions can always be substituted for each other in a more
complex expression - useful for simplification.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Propositional Equivalences (1.2) (cont.)

Normal or Canonical Forms


Unique representations of a proposition
Examples: Construct a simple proposition of two
variables which is true only when
P is true and Q is false: P  Q
 P is true and Q is true: P  Q
 P is true and Q is false or P is true and Q is true:
(P  Q) V (P  Q)

CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Propositional Equivalences (1.2) (cont.)

A disjunction of conjunctions where


every variable or its negation is represented once in each
conjunction (a minterm)
each minterms appears only once
Disjunctive Normal Form (DNF)

Important in switching theory, simplification in the design
of circuits.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Propositional Equivalences (1.2) (cont.)

Method: To find the minterms of the DNF.



Use the rows of the truth table where the proposition is 1
or True
If a zero appears under a variable, use the negation of
the propositional variable in the minterm
If a one appears, use the propositional variable.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Propositional Equivalences (1.2) (cont.)

Example: Find the DNF of (P V Q) R
P
Q
R
(P V Q) R
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
1
1
1
0
1
0
1
0
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Propositional Equivalences (1.2) (cont.)

There are 5 cases where the proposition is true, hence 5
minterms. Rows 1,2,3, 5 and 7 produce the following
disjunction of minterms:
(P V Q) R
 (P  Q  R) V (P  Q  R) V (P  Q  R)
V (P  Q  R) V (P  Q  R)

Note that you get a Conjunctive Normal Form (CNF) if
you negate a DNF and use DeMorgan’s Laws.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3)

A generalization of propositions - propositional
functions or predicates: propositions which contain
variables

Predicates become propositions once every variable
is bound- by

assigning it a value from the Universe of Discourse U
or

quantifying it
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Examples:

Let U = Z, the integers = {. . . -2, -1, 0 , 1, 2, 3, . . .}


Examples of propositions where x is assigned a value:




P(x): x > 0 is the predicate. It has no truth value until the
variable x is bound.
P(-3) is false,
P(0) is false,
P(3) is true.
The collection of integers for which P(x) is true are the
positive integers.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

P(y) V P(0) is not a proposition. The variable y
has not been bound. However, P(3) V P(0) is a
proposition which is true.

Let R be the three-variable predicate R(x, y z):
x+y=z

Find the truth value of
R(2, -1, 5), R(3, 4, 7), R(x, 3, z)
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Quantifiers

Universal
P(x) is true for every x in the universe of discourse.
Notation: universal quantifier
x P(x)
‘For all x, P(x)’, ‘For every x, P(x)’
The variable x is bound by the universal quantifier
producing a proposition.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Example: U = {1, 2, 3}
x P(x)  P(1)  P(2)  P(3)
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Quantifiers (cont.)

Existential

P(x) is true for some x in the universe of discourse.
Notation: existential quantifier
x P(x)
‘There is an x such that P(x),’ ‘For some x, P(x)’, ‘For
at least one x, P(x)’, ‘I can find an x such that P(x).’
Example: U={1,2,3}
x P(x)  P(1) V P(2) V P(3)
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Quantifiers (cont.)

Unique Existential
P(x) is true for one and only one x in the universe of
discourse.
Notation: unique existential quantifier
!x P(x)
‘There is a unique x such that P(x),’ ‘There is one and
only one x such that P(x),’ ‘One can find only one x
such that P(x).’
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Example: U = {1, 2, 3, 4}
P(1) P(2) P(3)
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
!xP(x)
0
1
1
0
1
0
0
0
How many
minterms are
in the DNF?
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)
REMEMBER!
A predicate is not a proposition until all variables
have been bound either by quantification or
assignment of a value!
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Equivalences involving the negation operator
(x P(x ))  x P(x)
(x P(x))  x P(x)

Distributing a negation operator across a quantifier
changes a universal to an existential and vice
versa.

(x P(x))  (P(x1)  P(x2)  …  P(xn))
 P(x1) V P(x2) V … V P(xn)
 x P(x)
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Multiple Quantifiers: read left to right . . .

Example: Let U = R, the real numbers,
P(x,y): xy= 0
x y P(x, y)
x y P(x, y)
x y P(x, y)
x y P(x, y)
The only one that is false is the first one.
What’s about the case when P(x,y) is the predicate
x/y=1?
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Multiple Quantifiers: read left to right . . .

Example: Let U = {1,2,3}. Find an expression equivalent
to x y P(x, y) where the variables are bound by
substitution instead:
Expand from inside out or outside in.
Outside in:
y P(1, y)  y P(2, y)  y P(3, y)
[P(1,1) V P(1,2) V P(1,3)] 
[P(2,1) V P(2,2) V P(2,3)] 
[P(3,1) V P(3,2) V P(3,3)]
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Converting from English (Can be very difficult!)
“Every student in this class has studied calculus”
transformed into:
“For every student in this class, that student has studied
calculus”
C(x): “x has studied calculus”
x C(x)
This is one way of converting from English!
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Multiple Quantifiers: read left to right . . . (cont.)

Example:
F(x): x is a fleegle
S(x): x is a snurd
T(x): x is a thingamabob
U={fleegles, snurds, thingamabobs}
(Note: the equivalent form using the existential quantifier is also
given)
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Everything is a fleegle
x F( x)
  (x F(x))

Nothing is a snurd.
x  S(x)
  (x S( x))

All fleegles are snurds.
x [F(x)S(x)]
 x [F(x) V S(x)]
 x  [F(x)  S(x)]
  (x [F(x) V S(x)])
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)



Some fleegles are thingamabobs.
x [F(x)  T(x)]
 (x [F(x) V T(x)])
No snurd is a thingamabob.
x [S(x) T(x)]
 (x [S(x )  T(x)])
If any fleegle is a snurd then it's also a thingamabob
x [(F(x)  S(x))  T(x)]
 (x [F(x)  S(x)  T( x)])
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Extra Definitions:



An assertion involving predicates is valid if it is true for
every universe of discourse.
An assertion involving predicates is satisfiable if there is a
universe and an interpretation for which the assertion is
true. Else it is unsatisfiable.
The scope of a quantifier is the part of an assertion in
which variables are bound by the quantifier
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Examples:
Valid: x S(x)  [x S( x)]
Not valid but satisfiable: x [F(x)  T(x)]
Not satisfiable: x [F(x)  F(x)]
Scope: x [F(x) V S( x)] vs. x [F(x)] V x [S(x)]
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Predicates & Quantifiers (1.3) (cont.)

Dangerous situations:

Commutativity of quantifiers
x y P(x, y) y x P( x, y)?
YES!
x y P(x, y)  y x P(x, y)?
NO!
DIFFERENT MEANING!

Distributivity of quantifiers over operators
x [P(x)  Q(x)]  x P( x)  x Q( x)?
YES!
x [P( x)  Q( x)] [x P(x)  x Q( x)]?
NO!
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
32
Sets (1.6)

A set is a collection or group of objects or elements
or members. (Cantor 1895)


A set is said to contain its elements.
There must be an underlying universal set U, either
specifically stated or understood.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
33
Sets (1.6) (cont.)

Notation:



list the elements between braces:
S = {a, b, c, d}={b, c, a, d, d}
(Note: listing an object more than once does not change the set.
Ordering means nothing.)
specification by predicates:
S= {x| P(x)},
S contains all the elements from U which make the predicate P
true.
brace notation with ellipses:
S = { . . . , -3, -2, -1},
the negative integers.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
34
Sets (1.6) (cont.)

Common Universal Sets





R = reals
N = natural numbers = {0,1, 2, 3, . . . }, the counting
numbers
Z = all integers = {. . , -3, -2, -1, 0, 1, 2, 3, 4, . .}
Z+ is the set of positive integers
Notation:
x is a member of S or x is an element of S:
x  S.
x is not an element of S:
x  S.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
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Sets (1.6) (cont.)

Subsets


Definition: The set A is a subset of the set B, denoted
A  B, iff
x [x  A  x  B]
Definition: The void set, the null set, the empty set,
denoted , is the set with no members.
Note: the assertion x   is always false. Hence
x [x    x  B]
is always true(vacuously). Therefore,  is a subset of
every set.
Note: A set B is always a subset of itself.
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
36
Sets (1.6) (cont.)



Definition: If A  B but A  B the we say A is a proper
subset of B, denoted A  B (in some texts).
Definition: The set of all subset of a set A, denoted P(A),
is called the power set of A.
Example: If A = {a, b} then
P(A) = {, {a}, {b}, {a,b}}
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
37
Sets (1.6) (cont.)

Definition: The number of (distinct) elements in A,
denoted |A|, is called the cardinality of A.
If the cardinality is a natural number (in N), then the
set is called finite, else infinite.

Example: A = {a, b},
|{a, b}| = 2,
|P({a, b})| = 4.
A is finite and so is P(A).
Useful Fact: |A|=n implies |P(A)| = 2n
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
38
Sets (1.6) (cont.)

N is infinite since |N| is not a natural number. It is called a transfinite
cardinal number.

Note: Sets can be both members and subsets of other sets.

Example:
A = {,{}}.
A has two elements and hence four subsets:
, {}, {{}}. {,{}}
Note that  is both a member of A and a subset of A!


Russell's paradox: Let S be the set of all sets which are not members
of themselves. Is S a member of itself?
Another paradox: Henry is a barber who shaves all people who do
not shave themselves. Does Henry shave himself?
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions
39
Sets (1.6) (cont.)

Definition: The Cartesian product of A with B, denoted
A x B, is the set of ordered pairs {<a, b> | a  A  b  B}
Notation: n
 Ai   a1 , a 2 ,...,a n  a i  Ai 
i 1
Note: The Cartesian product of anything with  is . (why?)


Example:
A = {a,b}, B = {1, 2, 3}
AxB = {<a, 1>, <a, 2>, <a, 3>, <b, 1>, <b, 2>, <b, 3>}
What is BxA? AxBxA?
If |A| = m and |B| = n, what is |AxB|?
CS 210, Ch.1 (part 2): The foundations: Logic & Proof, Sets, and Functions