Logic, contrapositive, converse, Discrete Mathematics, conjunction, negation

Discrete
Structures
or
Mathematics
Muhammad Nawaz, (PhD, UK)
Assistant Professor (Multimedia Systems)
PhD and MS-Computer Science Programs Coordinator
Centre for Excellence in Information Technology
IMSciences Peshawar- Pakistan

Introduction
• Discrete mathematics is the part of mathematics devoted to the study
of discrete objects.
• Here “Discrete” means consisting of unconnected elements.
• Examples
• How many ways are there to choose a valid password on a computer
system?
• What is the probability of winning a lottery?
• Is there a link between two computers in a network?
• How can I identify spam email messages?

• How can I encrypt a messages so that no unintended recipient can
read it?
• What is the shortest path between two cities using a transportation
system?
• How can a list of integers be sorted so that the integers are in
increasing order?
• How many steps are required to do such a sorting?
• How can it be proved that a sorting algorithm correctly sorts a list?
• How can a circuit that adds two integers be designed?
• How many valid Internet addresses are there?

• More generally, discrete mathematics is used whenever objects are
counted, when relationships between ﬁnite (or countable) sets are
studied, and when processes involving a ﬁnite number of steps are
analyzed.

Why Study Discrete Mathematics?
• There are several important reasons for studying discrete
mathematics.
• First, through this course you can develop your mathematical
maturity: that is, your ability to understand and create mathematical
arguments.
• Second, discrete mathematics is the gate way to more advanced
courses in all parts of the mathematical sciences i.e. data structures,
algorithms, database theory, automata theory, formal languages,
compiler theory, computer security, and operating systems.

• Also, discrete mathematics contains the necessary mathematical
background for solving problems in operations research ( including
many discrete optimization techniques), chemistry, engineering,
biology, and soon.

Logic?????
• Logic is the study of the principles and methods that distinguish
between a valid and invalid argument.

Propositions
• A proposition is a declarative sentence (that is, a sentence that
declares a fact) that is either true or false, but not both.
• Examples
1. Islamabad is the capital of Pakistan.
2. Madras is the capital of India.
3. 1+1=2.
4. 2+2=3.
• Note that sentences 1 and 3 are true but 2 an 4 are false but in any
case all the four sentences are proposition according to the definition

• Some sentences that are not propositions are given as follows.
1. What time is it?
2. Read this carefully.
3. x+1=2.
4. x+y=z.
• Sentences 1 and 2 are not propositions because they are not declarative
sentences. Sentences 3 and 4 are not propositions because they are
neither true nor false.
• Note that each of sentences 3 and 4 can be turned into a proposition if we
assign values to the variables.

Proposition Variables
• Propositions are denoted by letters i.e. p, q, r, s.
• The truth value of proposition is denoted by T and False values are
denoted by F.

Atomic Propositions
• Atomic proposition is the proposition that cannot be divided into
more simpler propositions.
• For example
• “Dr. Sajid Anwar is the assistant professor in IMS.
• 50>40
• Ali is rich

Molecular Proposition
• When atomic propositions are combined with the help of connective,
molecular propositions are formed. (and, or, not).
• For example consider the following propositions
• p: Pakistan is situated in South Asia
• q: Islamabad is the capital of Pakistan
• By using connective “and” a molecular proposition can be formed as;
• r: Pakistan is situated in South Asia and Islamabad is the capital of
Pakistan.

Truth Table
• A truth table displays the relationship between the truth values of
propositions.
• It is constructed from simpler propositions.
• Let see how to construct a truth tables for different forms of
propositions???

Negation
• Let p be a proposition. The negation of p, denoted by ~p (also
denoted by p), is the statement
“It is not the case that p.”
• The proposition ~p is read “not p.” The truth value of the negation of
p, ~p, is the opposite of the truth value of p.
• For example
• “Today is Friday”
• The negation of the given sentence would be
• “Today is not Friday”

• “Ali’s smart phone has at least 32GB of memory”
• The negation of this sentence would be
• “Ali’s smart phone does not have at least 32GB of memory”
• Or simply
• “Ali’s smart phone has less than 32GB of memory”

Truth table for Negation
• Truth table for negation is given
in the table shown.
• T represents true value and F
represents false value.

Conjunction ()
• If p and q are statements, then
the conjunction of p and q is “p
and q”, denoted as “p  q”.
• It is true when, and only when,
both p and q are true. If either p
or q is false, or if both are false,
p  q is false.
• Truth table for conjunction can
be shown as;

Disjunction ()
• If p & q are statements, then the
disjunction of p and q is “p or
q”, denoted as “p  q”.
• It is true when at least one of p
or q is true and is false only
when both p and q are false.
• Truth table for disjunction is
shown as;

Conditional statements or Implications
• Let p and q be propositions. The
conditional statement p → q is the
proposition “if p, then q.” The
conditional statement p → q is
false when p is true and q is false,
and true otherwise.
• In the conditional statement p → q,
p is called the hypothesis (or
antecedent or premise) and q is
called the conclusion (or
consequence).
• Truth table for implication is shown
as;
p q p q
T T T
T F F
F T T
F F T

Implication Examples
• If there is flood then the crops will destroy.
• Hassan will pass the exam if he studies hard.
• If it rains then the streets get wet.
• If It snows then we will go on skiing.
• “If 1 = 1, then 3 = 3.” TRUE
• “If 1 = 1, then 2 = 3.” FALSE
• “If 1 = 0, then 3 = 3.” TRUE
• “If 1 = 2, then 2 = 3.” TRUE
• “If 1 = 1, then 1 = 2 and 2 = 3.” FALSE
• “If 1 = 3 or 1 = 2 then 3 = 3.” TRUE

Inverse of a Conditional Statement
• The inverse of the conditional statement p  q is ~p  ~q
• A conditional and its inverse are not equivalent as could be seen from
the truth table.

Examples
1. If today is Friday, then 2 + 3 = 5.
If today is not Friday, then 2 + 3  5.
2. If it snows today, I will ski tomorrow.
If it does not snow today I will not ski tomorrow.
3. If P is a square, then P is a rectangle.
If P is not a square then P is not a rectangle.
4. If my car is in the repair shop, then I cannot get to class.
If my car is not in the repair shop, then I shall get to the class.

Converse of a conditional statement
• The converse of the conditional
statement p  q is q p
• A conditional and its converse
are not equivalent i.e.,  is not
a commutative operator.

Examples
If 2 + 3 = 5, then today is Friday.
I will ski tomorrow only if it snows today.
If P is a rectangle then P is a square.
If I cannot get to the class, then my car is in the repair shop.

Contrapositive of a conditional statement
• The contrapositive of the conditional statement p  q is ~ q  ~ p
• A conditional and its contrapositive are equivalent.
If 2 + 3  5, then today is not Friday.
I will not ski tomorrow only if it does not snow today.
If P is not a rectangle then P is not a square.
If I get to the class, then my car is not in the repair shop.

Biconditional
• If p and q are statement
variables, the biconditional of p
and q is
• “p if and only if, q” and is
denoted p  q. if and only if
abbreviated iff.
• The double headed arrow " "
is the biconditional operator.
• Truth table for biconditional
statement is shown as;

Examples
• True or false?
1. “1+1 = 3 if and only if earth is flat”
TRUE
2. “Sky is blue iff 1 = 0”
FALSE
3. “Milk is white iff birds lay eggs”
TRUE
4. “33 is divisible by 4 if and only if horse has four legs”
FALSE

Compound Propositions
• A string consisting of variables, parenthesis and connectives is called a
compound proposition.
• Example
• ~(p  q), ~(p Ʌ q), (p(p  q  r))
Note: “2n” is used to create a truth table for the given expression.
Where n is the number of variable used in the expression.

Tautology
• A tautology is a statement form
that is always true regardless of
the truth values of the
statement variables.
• A tautology is represented by
the symbol “T”.
• EXAMPLE: The statement form p
 ~ p is tautology

Contradiction
• A contradiction is a statement
form that is always false
regardless of the truth values of
the statement variables.
• A contradiction is represented
by the symbol “c”.
• EXAMPLE:
• The statement form p  ~ p is a
contradiction.
p ~ p p  ~ p
T F F
F T F

Exclusive Or
• When or is used in its exclusive sense, the statement “p or q” means
“p or q but not both” or “p or q and not p and q” which translates
into symbols as:
• (p  q)  ~ (p  q)
• Which is abbreviated as:
• p  q
• or p XOR q

• Basically
• p  q ≡ (p Ʌ ~q) v (~p Ʌ q)
• ≡ [p Ʌ ~q) v ~p] Ʌ [(p Ʌ ~q) v q]
• ≡ (p v q) Ʌ ~(p Ʌ q)
• ≡ (p v q) Ʌ (~p v ~q)
•
p q pq pq ~ (pq) (pq)  ~ (pq)
T T T T F F
T F T F T T
F T T F T T
F F F F T F
TRUTH TABLE FOR EXCLUSIVE OR:

Logical Equivalence
• Two compound propositions A, B are regarded as logically equivalent,
if they have the same truth values.
OR
• Two compound propositions A, B are logically equivalent if and only if
A  B is a tautology.
• Examples

• p Ʌ T  p Laws of identity
• p v F  p
• p Ʌ p  p Idempotent Law
• p v p  p
• ~(~p)  p Double Negative
• p v q  q v p
• p Ʌ q  q Ʌ p
• Commutative Law
• ((p v q) Ʌ r)  ((p v (q v r))
• ((p v (q Ʌ r)  ((p v q) Ʌ (p v r))
• Distributive Laws
• ((p v q) v r)  ((p v (q v r))
• ((p Ʌ q) Ʌ r)  ((p Ʌ (q Ʌ r))
• Associative Laws
• ~(p v q)  ~p Ʌ ~q
• ~(p Ʌ q)  ~p v ~q
• Demargan’s Laws
• p  q  ~q  ~p Contrapositive

Applying Laws Of Logic
• Using law of logic, simplify the statement form
• p  [~(~p  q)]
• Solution:
• p  [~(~p  q)]  p  [~(~p)  (~q)] DeMorgan’s Law
• p  [p(~q)] Double Negative Law
•  [p  p](~q) Associative Law for 
• p  (~q) Indempotent Law

• Using Laws of Logic, verify the logical equivalence
• ~ (~ p  q)  (p  q) p
• Solution:
• ~(~p  q)  (pq)  (~(~p)  ~q) (p  q) DeMorgan’s Law
•  (p  ~q)  (p  q) Double Negative Law
 p  (~q  q) Distributive Law
•  p  c Negation Law
•  p Identity Law

Translating From English To Symbols
• Let p = “It is hot”, and q = “It is sunny”
• SENTENCE SYMBOLIC FORM
• It is not hot. ~ p
• It is hot and sunny. p q
• It is hot or sunny. p  q
• It is not hot but sunny. ~ p q
• It is neither hot nor sunny. ~ p  ~ q

• p = “Islamabad is the capital of Pakistan”
• q = “17 is divisible by 3”
• p Ʌ q = “Islamabad is the capital of Pakistan and 17 is divisible by 3”
• p v q = “Islamabad is the capital of Pakistan or 17 is divisible by 3”
• ~p = “It is not the case that Islamabad is the capital of Pakistan” or
simply
• “Islamabad is not the capital of Pakistan”

• Let h = “Zia is healthy”
• w = “Zia is wealthy”
• s = “Zia is wise”
• Translate the compound statements to symbolic form:
1. Zia is healthy and wealthy but not wise. (h  w)  (~s)
2. Zia is not wealthy but he is healthy and wise. ~w  (h  s)
3. Zia is neither healthy, wealthy nor wise. ~h  ~w  ~s

Translating From Symbols To English
• Let m = “Ali is good in Mathematics”
• c = “Ali is a Computer Science student”
• Translate the following statement forms into plain English:
• ~ c Ali is not a Computer Science student
• c  m Ali is a Computer Science student or good in Maths.
• m  ~c Ali is good in Maths but not a Computer Science
student
• A convenient method for analyzing a compound statement is to make
a truth table for it.

Rules of Inference
• In propositional logic there are certain statements that are accepted
as axioms- statements that do not need to be proved. An axiom is
therefore a statement whose truth is self evident.
• On the other hand a theorem is a statement that is shown to be true.
It can be demonstrated that a theorem is true with the sequence of
statements that form an argument, called a proof.
• To construct proofs, methods are needed to derive new statements
from the old ones. Here axioms are used.
• The rules of inference tie together the step of a proof.

Rules of Inference
• Let P an Q are two compound
propositions. P logically implies
Q (i.e. P => Q) if and only if P 
Q is tautology.
• Example
• p  (p v q) Addition
• p Ʌ q  p Simplification
• (p c)  ~p Absurdity
• (p Ʌ (p  q))  q Modus ponens
• ((p  q) Ʌ ~q => ~p
• Modus tollens
• ((p q) Ʌ (q  r)) => p  r
• Transitivity of Implication
• ((p  q) Ʌ (q  r)) (p  r)
• Hypothetical syllogism
• ((p v q) Ʌ ~p)  q
• Disjunctive syllogism

Argument
• An argument is a list of statements called premises (or assumptions or
hypotheses) followed by a statement called the conclusion.
• P1 Premise
• P2 Premise
• P3 Premise
• . . . . .. . . . .
• Pn Premise
• ______________
• C Conclusion
• NOTE The symbol  read “therefore,” is normally placed just before the
conclusion.

Valid argument
• An argument is valid if the
conclusion is true when all the
premises are true.
• Alternatively, an argument is
valid if conjunction of its
premises imply conclusion. That
is (P1 P2  P3  . . .  Pn)  C
is a tautology.
EXAMPLE:
Show that the following argument form is valid:
pq
p
 q
SOLUTION
premises conclusion
FFTFF
TFTTF
FTFFT
TTTTT
qppqqp
critical row

Invalid argument
• An argument is invalid if the
conclusion is false when all the
premises are true.
• Alternatively, an argument is
invalid if conjunction of its
premises does not imply
conclusion.
• EXAMPLE Show that the
following argument form is invalid:
• p  q
• q
•  p
SOLUTION premises conclusion
FFTFF
FTTTF
TFFFT
TTTTT
pqpqqp
critical row

Example
• Use truth table to determine the
argument form
• p  q
• p  ~q
• p  r
•  r
• is valid or invalid.
premises conclusion
FTTFFFF
TTTFTFF
FTTTFTF
TTTTTTF
FFTTFFT
TTTTTFT
FFFTFTT
TTFTTTT
rprp~qpqrqp
Critical rows

Word Form Example
• If Tariq is not on team A, then Hameed is
on team B.
• If Hameed is not on team B, then Tariq is
on team A. Tariq is not on team A or
Hameed is not on team B.
• SOLUTION
• Let
• p = Tariq is on team A
• q = Hameed is on team B
• Then the argument is
• ~ p  q
• ~ q  p
•  ~ p  ~ q
p q ~p  q ~p  q ~p ~q
T T T T F
T F T T T
F T T T T
F F F F T

• Argument is invalid. Because there are three critical rows ( Remember
that the critical rows are those rows where the premises have truth
value T) and in the first critical row conclusion has truth value F. (Also
remember that we say an argument is valid if in all critical rows
conclusion has truth value T).

• If at least one of these two
numbers is divisible by 6, then
the product of these two
numbers is divisible by 6.
• Neither of these two numbers is
divisible by 6.
•  The product of these two
numbers is not divisible by 6.
• SOLUTION
• Let p = at least one of these
two numbers is divisible by 6.
• q = product of these two
numbers is divisible by 6.
• Then the argument become in
these symbols
• p  q
• ~ p
•  ~ q

p q p  q ~p ~q
T T T F F
T F F F T
F T T T F
F F T T T
• Here there are two critical rows
the 3rd and 4th rows. The
conclusion of the third row is F.
This shows that the given
argument is invalid.

• If I got an Eid bonus, I’ll buy a
stereo.
• If I sell my motorcycle, I’ll buy a
stereo.
•  If I get an Eid bonus or I sell my
motorcycle, then I’ll buy a stereo.
• SOLUTION:
• Let
• e = I got an Eid bonus
• s = I’ll buy a stereo
• m = I sell my motorcycle
• The argument is
• e  s
• m  s
• e  m  s

e s me sm s emem s
T T T T T T T
T T F T T T T
T F T F F T F
T F F F T T F
F T T T T T T
F T F T T F T
F F T T F T F
F F F T T F T
• The argument is valid. Because
there are five critical rows (
Remember that the critical rows
are those rows where the
premises have truth value T) and
in all critical row conclusion has
truth value T. (Also remember
that we say an argument is valid
if in all critical rows conclusion
has truth value T).

• An interesting teacher keeps me
awake. I stay awake in Discrete
Mathematics class. Therefore, my
Discrete Mathematics teacher is
interesting.
• Solution:
• t: my teacher is interesting
a: I stay awake
• m: I am in Discrete Mathematics
class the argument to be tested is
• t  a,
• a  m
•  m  t
t a m t  a a  m m  t
T T T T T T
T T F T F F
T F T F F T
T F F F F F
F T T T T F
F T F T F F
F F T T F F
F F F T F F

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