Discrete Mathematics
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Transcript of Discrete Mathematics
Discrete Mathematics
Logic
Propositions A proposition is a statement or sentence
that can be determined to be either true or false (but no both).
Examples: The only positive integers that divide 7 are 1 and
7 itself. Buy two tickets for Friday concert. Earth is the only planet in the universe that
contains life.
Example Use variable to represent propositions
P: 1+1=3
P: It is raining outside
P: Today is Tuesday
ConnectivesIf p and q are propositions, new compound
propositions can be formed by using connectives
Most common connectives: Conjunction ^ Disjunction Negation ~ Exclusive-OR v Condition Bi-Condition
Example P: It is raining Q: It is cold
Form a new compound statement by combining these two statements
P Q : It is raining and it is cold P Q : It is raining or it is cold
Truth table of conjunction The truth values of compound propositions
can be described by truth tables. Truth table of conjunction
P Q is true only when both Pp and Q are true.
P Q P QT T T
T F F
F T F
F F F
Example
Let P = “A decade is 10 years” Let Q = “A millennium is 100 years” P Q = “A decade is 10 years” and “A
millennium is 100 years” If P is true and Q is false then conjunction
is false
Truth table of disjunction The truth table of disjunction is
p q is false only when both p and q are false Example: p = "John is a programmer", q = "Mary is a lawyer" p v q = "John is a programmer or Mary is a lawyer"
P Q P QT T T
T F T
F T T
F F F
Negation Negation of P: in symbols ~P
~P is false when P is true, ~P is true when P is false Example, P : "John is a programmer" ~P = "John is not a programmer"
P ~P
T F
F T
E.g
P: Paris is the capital of England ~P: Paris is not capital of England
Exclusive disjunction “Either P or Q” (but not both), in symbols P
Q
P Q is true only when P is true and Q is false, or P is false and Q is true. Example: p = "John is programmer, q = "Mary is a lawyer" p v q = "Either John is a programmer or Mary is a lawyer"
P Q P v Q
T T F
T F T
F T T
F F F
More compound statements
Let p, q, r be simple statements We can form other compound statements,
such as (pq)^r p(q^r) (~p)(~q) (pq)^(~r) and many others…
Example: truth table of (P Q)RP Q R (P Q) R
T T T T
T T F F
T F T T
T F F F
F T T T
F T F F
F F T F
F F F F
Conditional propositions
A conditional proposition is of the form “If P then Q” In symbols: P Q Example:
P = " A bottle contains acid" Q = “A bottle has a label” P Q = “If a bottle contains acid then it has a
label "
Truth table of P Q
P Q is true when both p and q are true or when P is false
P Q P Q
T T T
T F F
F T T
F F T
Example If the mathematics department gets an
additional $40,000 then it will hire one new faculty member.
Let P: The Mathematics Department gets an additional $40,000 and Q: The mathematics Department will hire one new faculty member.
Hypothesis and conclusion
In a conditional proposition P Q,
P is called the hypothesis
Q is called the conclusion
Example For all real number x if x > 0 then x2 > 0
For example x=3 , 3 > 0 then 32 > 0 both are true.
x=-2 , -2 > 0 is false but -22 > 0
Logical equivalence Two propositions are said to be logically equivalent if their truth tables are identical.
Example: ~P Q is logically equivalent to P Q
P Q ~P Q P Q
T T T T
T F F F
F T T T
F F T T
Converse The converse of p q is q p
These two propositions
are not logically equivalent
p
q p q q p
T T T T
T F F T
F T T F
F F T T
Contrapositive The contrapositive of the proposition p q is ~q
~p.
They are logically equivalent.
p q p q ~q ~p
T T T T
T F F F
F T T T
F F T T
Bi-Conditional The double implication “p if and only if q” is
defined in symbols as p q
p q is logically equivalent to (p q)^(q p)
p q p q (p q) ^ (q p)
T T T T
T F F F
F T F F
F F T T
Tautology A proposition is a tautology if its truth table
contains only true values for every case Example: p p v q
p q p p v q
T T T
T F T
F T T
F F T
Contradiction A proposition is a tautology if its truth table
contains only false values for every case Example: p ^ ~p
p p ^ (~p)
T F
F F
De Morgan’s laws for logic
The following pairs of propositions are logically equivalent:
~ (p q) and (~p)^(~q) ~ (p ^ q) and (~p) (~q)
Quantifiers
A propositional function P(x) is a statement involving a variable x
For example: P(x): 2x is an even integer
x is an element of a set D For example, x is an element of the set of integers
D is called the domain of P(x)
For every and for some
Most statements in mathematics and computer science use terms such as for every and for some.
For example: For every triangle T, the sum of the angles of T
is 180 degrees. For every integer n, n is less than p, for some
prime number p.
Universal quantifier
One can write P(x) for every x in a domain D In symbols: x P(x)
is called the universal quantifier
Inference Or Deduction An argument is a sequence of propositions
written as below : p1, p2 , p3 , p4 , p5 ,,…, , pn, therefore q.
(if p1v p2v,.., pn then q) or (if p1 ^p2 ^,.., pn then q)
the propositions p1, .., pn, are called hypotheses (premises) and proposition q is called conclusion (consequent).
The above argument is valid if the consequent can be proven from the premises.
Validity can be shown using truth table or by using theorems / rules.
Rules of Inference Addition p , therefore p v q
(read as if we know p is true, therefore we know p or q is true)
Simplification
p ^ q , therefore p Conjunction
p , q , therefore p ^ q
Rules of Inference (contd..) Modus Ponens p q , p, therefore q Modus Tollens ~ q, p q , therefore ~p Hypothetical Syllogism
p q, q r, therefore p r Disjunctive Syllogism p v q , ~ p, therefore q Resolution
p v q, ~ p v r , therefore q v r
ExampleState which rule of inference is the basis of the
following argument :“ It is below freezing now. Therefore it is either
below freezing or raining now.”- addition rule
“It is below freezing and raining now. Therefore, it is below freezing now. “- simplification rule
“Rules : If it snows today, then we will go skiing”Fact : it snows today. Conclude: We will go skiing”
- modus ponens
Problem
Show that the hypotheses : “ It is not sunny this afternoon and it is colder than
yesterday.”Will conclude :“We will be home by sunset “Given the following fact :“We will go swimming only if it is sunny. If we do
not go swimming, then we will take a canoe trip. If we take a canoe trip we will be home by sunset”