Applications of Propositional Logic in Logical Structures

LOGICAL
STRUCTURES
Discrete Structures 1
Lecture 1

Discrete Mathematics and Its
Applications by Kenneth H. Rosen 2
LOGIC
 is the study of reasoning.
 is specially concerned with whether
reasoning is correct
 focuses on the relationship among
statements as opposed to the content of
any particular statement

PROPOSITION
 A proposition is a declarative sentence
that is either true or false, but not both.
 Example:
Manila is the capital of the Philippines.
1 + 1 = 3.
2 < 5.
Today is Friday.

PROPOSITION
 Consider the following sentences:
 What time is it?
 Read this carefully.
 x + 1 = 2.
 x + y = z.
 Sentences 1&2 are not propositions because
they are not declarative sentences. Sentences
3&4 are not propositions because they are
neither true nor false, since the variables in
these sentences have not been assigned
values.

Proposition
Prime Proposition- simple
statement that express a single
complete thought, and the whole
truth value is true
Ex.
1.The sun is shining.
2.I walk to work.

Compound Proposition – are
propositions that are made of two or more
propostions and whose truth value is true
or false, but never both.
Ex.
1.The sun is shining and I work to work.
Socrates is a man or Socrates is a dog

PROPOSITION
 Conventional letters used to denote
propositions are p, q, r, s,…
 The truth value of a proposition is true,
denoted by T, if it is true and false,
denoted by F, if it is a false proposition.

NEGATION
 Let p be a proposition. The statement
“It is not the case that p”
is another proposition called the negation
of p.
 The negation of p is denoted by p. The
proposition is read “not p.”

NEGATION
 Example:
p: Today is Friday.
p: It is not the case that today is
Friday. Or Today is not Friday.
p: Michael’s PC runs Linux
p: It is not the case that Michael’s PC runs
Linux

NEGATION
 Truth Table
p p
T F
F T

Exercise
Give the negation of the following
propositions:
1.John missed the final examinations.
2.Mary is a vegetarian.
3.1 + 5 = 9.
4.4 > 0.

CONJUNCTION
 Let p and q be propositions. The
proposition “p and q”, denoted p  q, is the
proposition that is true when both p and q
are true and is false otherwise. The
proposition p  q is called the conjunction
of p and q.

CONJUNCTION
 Example:
p: Today is Friday.
q: It is raining today.
p  q: Today is Friday and it is raining
today.

CONJUNCTION
 Truth Table:
p q p  q
T T T
T F F
F T F
F F F

Exercise
 Let p and q be the propositions
p: It is below freezing.
q: It is snowing.
Write these propositions using p and q and
logical connectives.
a. It is below freezing and snowing.
b. It is below freezing but not snowing.
c. It is not below freezing and it is not snowing.

Exercise
 Calculate the truth values:
a. (1>0)  (2<1)
b. (0<1)  (1<2)
 Write out the truth tables for
a. p  (q)
b. (p  q)  r

DISJUNCTION
proposition “p or q”, denoted p  q, is the
proposition that is false when p and q are
both false and true otherwise. The
proposition p  q is called the disjunction of
p and q.

DISJUNCTION
 Example 1:
p: Today is Friday.
q: It is raining today.
p  q: Today is Friday or it is raining today.
 Example 2:
r: I want to eat chicken.
s: I want to eat spaghetti.
r  s: I want to eat chicken or spaghetti.

DISJUNCTION
 Truth Table:
p q p  q
T T T
T F T
F T T
F F F

Exercise
 Let p and q be the propositions
p: The election is decided.
q: The votes have been counted.
Express each of these compound
propositions as an English sentence:
a. p  q
b. q  (p  q)

Exercise
 Calculate the truth values (assuming p =
T, q = F, r = F)
a.  p  (q  r)
b. (p  q)  (p  r)
 Write out the truth tables for
a. p  (p  q)
b. (q  r)  (p  q)

EXCLUSIVE OR
 Let p and q be propositions. The exclusive
or of p and q, denoted by pq, is the
proposition that is true when exactly one of
p and q is true and false otherwise.

EXCLUSIVE OR
 Example:
p: Students who have taken calculus
can enroll in this class.
q: Students who have taken computer
science can enroll in this class.
p  q: Students who have taken
calculus or computer science,
but not both, can enroll in this class.

EXCLUSIVE OR
 Truth Table:
p q p  q
T T F
T F T
F T T
F F F

IMPLICATION
implication p  q is the proposition that is
false when p is true and q is false, and
true otherwise. In this implication p is
called the hypothesis (or antecedent or
premise) and q is called the conclusion (or
consequent).

IMPLICATION
 Example:
p: I am elected.
q: I will lower taxes.
p  q: If I am elected, then I will
lower taxes.

IMPLICATION
 Ways to express implication:
“if p, then q” “p implies q”
“if p, q” “p only if q”
“q if p” “q whenever p”
“q when p” “q is necessary for p”
“q follows from p”“p is sufficient for q”
“a sufficient condition for q is p”
“a necessary condition for p is q”

Exercise
 State each statement in the form “if p, then
q” in English:
a. It snows whenever the wind blows from the
northeast.
b. The apple trees will bloom if it stays warm for a
week.
c. It is necessary to walk 8 miles to get to the top
of Long’s Peak.
d. To get tenure as a professor, it is sufficient to be
world-famous.
e. Your guarantee is good only if you bought your
CD player less than 90 days ago.

IMPLICATION
 Truth Table:
p q p  q
T T T
T F F
F T T
F F T

IMPLICATION
 q  p is called the converse of p  q.
 q  p is called the contrapositive of
pq.
 p  q is called the inverse of pq

IMPLICATION
 What is the contrapositive, converse, and
inverse of the ff. propositions
a. The home team wins whenever it is raining.
b. That you get a job implies that you had the best
credentials.
c. To be a citizen of this country, it is sufficient that
you were born in the Philippines.
d. I will remember to send you the address only if
you send me an e-mail message.

IMPLICATION
 The contrapositive, q  p, of an
implication p  q has the same truth value
(or equivalent) as p  q.
 Neither the converse nor the inverse have
the same truth value as p  q for all
possible truth values of p and q.

Exercise
 Determine whether these implications
are true or false:
a. If 1+1=2, then 2+2=5.
b. If 1+1=3, then 2+2=4.
c. If pigs can fly, then 1+1=3.
d. If 1+1=3, then pigs can fly.

Exercise
 Construct the truth table for each of
these compound propositions:
a. (p  q)  (p q)
b. (p  q)  (p  q)
c. p  (q  r)
d. (p  q)  (p  r)

BICONDITIONAL
biconditional p  q is the proposition that
is true when p and q have the same truth
values, and is false otherwise.

BICONDITIONAL
 Example:
p: You can take the flight.
q: You buy a ticket.
p  q: You can take the flight if and
only if you buy a ticket.

BICONDITIONAL
 Ways to express biconditional statement:
“p if and only if q”
“p is necessary and sufficient for q”
“if p then q, and conversely”
“p if q”

BICONDITIONAL
 Truth Table:
p q p  q
T T T
T F F
F T F
F F T

Exercise
 Express each proposition in the form “p if
and only if q” in English.
a. For you to get a 5.0 in this course, it is
necessary and sufficient that you learn how to
solve discrete mathematics problems.
b. It rains if it is a weekend day, and it is a
weekend day if it rains.
c. You can see the wizard only if the wizard is not
in, and the wizard is not in only if you can see
him.

Precedence of Logical Operators
Operator Precedence
 1


2
3


4
5

Definitions
 A compound proposition that is always
true, no matter what the truth values of the
propositions that occur in it, is called a
tautology.
 A compound proposition that is always
false is called a contradiction.
 A proposition that is neither a tautology
nor a contradiction is called a contingency.

Tautologies and Contradictions
 Examples of Tautology and Contradiction:
p  p and p  p
p p p  p p  p
T
F
F
T
T
T
F
F

LOGICAL EQUIVALENCES
 The propositions p and q are called
logically equivalent if p  q is a tautology.
The notation p  q denotes that p and q
are logically equivalent.

LOGICAL EQUIVALENCES
 Example: Show that (p  q) and p  q are
logically equivalent. (one of De Morgan’s
laws)
p q p  q (p  q) p q p  q
T T
T F
F T
F F
T
T
T
F
F
F
F
T
F
F
T
T
F
T
F
T
F
F
F
T

Exercise
 Show that p  q and p  q are logically
equivalent.
 Show that p  (q  r) and (p  q)(p  r) are
logically equivalent. Identify the Law.

Logical Equivalences
Equivalence Name
p  T  p
p  F  p
Identity laws
p  T  T
p  F  F
Domination laws
p  p  p
p  p  p
Idempotent laws
(p)  p Double negation law

Equivalence Name
p  q  q  p
p  q  q  p
Commutative laws
(p  q)  r p  (q  r)
(p  q)  r  p  (q  r)
Associative laws
p  (q  r)  (p  q)  (p  r)
p  (q  r)  (p  q)  (p  r)
Distributive laws
(p  q)  p  q
(p  q)  p  q
De Morgan’s laws

p  (p  q)  p
p  (p  q)  p
Absorption laws
p  p  T
p  p  F
Negation laws

Logical Equivalences Involving
Implications
 p  q  p  q
 p  q  q  p
 p  q  p  q
 p  q  (p   q)
 (p  q)  p   q
 (p  q)  (p  r)  p  (q  r)
 (p  r)  (q  r)  (p  q)  r
 (p  q)  (p  r)  p  (q  r)
 (p  r)  (q  r)  (p  q)  r

Logical Equivalences Involving
Biconditionals
 p  q  (p  q)  (q  p)
 p  q  p   q
 p  q  (p  q)  (p  q)
 (p  q)  p  q

Equational Reasoning
 Show that (p  q)  (p  q) is a tautology.
 Show that (p  (p  q)) and p  q are
logically equivalent.
 Show that (p  q)  ( p  q) is a tautology.
 Show that (p  q)  p and (q  p) are
logically equivalent.

Applications of Propositional Logic in Logical Structures

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