Post on 02-Jan-2016
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Discrete Structures – CS2300
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Text
Discrete Mathematics and Its Applications
Kenneth H. Rosen (7th Edition)
Chapter 1
The Foundations: Logic and Proofs
About This Course
• The Conceptual Foundation of Computer Science
• Prerequisite for CS 3240 (Theory of Computation)
• Applied Mathematics Course
Continuous vs. Discrete Math
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Continuous Discrete
Sliding down a slidePouring water
Length of ropeCrawling slug
Adding milkGrade point average
Climbing up stairsStacking ice cubesNumber of knotsHopping rabbitAdding eggsCalculus grade
Discrete Solutions
• How many ways are there to choose a valid password?
• What is the probability of winning the lottery?• Is there a path linking two particular computers
in a network?• What is the shortest path between two
destinations using a transportation system?• How many valid Internet addresses are there?
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Chapter 1 Objective
“In this chapter we will explain what makes up a correct mathematical [logical] argument and introduce tools to construct these arguments.”
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Sections 1.1, 1.2
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Logic
Propositional Logic
Propositions
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A proposition is a statement that is either true or false, but not both.Today is Tuesday.
Six is a prime number.
Count is less than ten.
7<5
Consider this statement.
Compound Propositions
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Compound propositions are formed from existing propositions using logical operatorsToday is Wednesday and it is snowing outside.
12 is not a prime number.
Negation of a Proposition
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T
F
F
T
NOT !P P
Negation of a Proposition
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repeat{…}until(feof(my_file));
while (!feof(my_file)){…}
Disjunction of Two Propositions
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T T
T F
F T
F F
T
T
T
F
OR ||qp p q
Disjunction of Two Propositions
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repeat{ …}until(count>10 || feof(myfile));
if(choice==PAUSE || choice ==STOP) ...
Conjunction of Two Propositions
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T T
T F
F T
F F
T
F
F
F
AND &&p q qp
Conjunction of Two Propositions
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while(!feof(a_file) && index<SIZE){ …}
if(!done && time_left) ...
Exclusive-OR of Two Propositions
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T T
T F
F T
F F
F
T
T
F
Exactlyone ofthem istrue.
p q qp ^
“but not both”
Implication
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T T
T F
F T
F F
T
F
T
T
p is called thehypothesis and q is theconclusion
p q qp
Implication (“Conditional”)
• “if p, then q”
• “p implies q”
• “if p,q”
• “p only if q”
• “p is sufficient for q”
• “q if p”
• “q whenever p”
• “q is necessary for p”
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T T
T F
F T
F F
T
F
T
T
p q qp
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q whenever p
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T T
T F
F T
F F
T
F
T
T
Suppose that the proposition is true. Then, q is true whenever p is true.
p q qp
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p is sufficient for q
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T T
T F
F T
F F
T
F
T
T
Suppose that the proposition is true. Then, to guarantee that q is true it is sufficient to say that p is true.
p q qp
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Converse of an Implication
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T T
T F
F T
F F
T
F
T
T
T
T
F
T
p q qp AndConversely
qp
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Example of Converse
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If it stays warm for a week, the apple trees will bloom.If the apple trees bloom, it will be warm for a week.
If x is even then x2 is even.
If x2 is even then x is even.
Contrapositive of an Implication
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T T
T F
F T
F F
T
F
T
T
T
F
T
F F
F T
T F
T T T
p qqp pq pq
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Examples of Contrapositive
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If it snows tonight, then I will stay at home.
If I do not stay at home, then it didn’t snow tonight.
If x is odd then x2 is odd.
If x2 is not odd then x is not odd.
If x2 is even then x is even.
Biconditional
T T
T F
F T
F F
T
F
T
T
T
T
F
T
T
F
F
T
p qpq qp )()( qpqp
qp
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Biconditional
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pif and only if q p iff q
qp )()( qpqp
Bitwise operators
1101 10011110 01001100 0000
AND1101 10011110 01001111 1101
OR
1101 10011110 01000011 1101
XOR
a&b a|b
a^b
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t01_1_009.jpg
Tautology
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Tautology - a compound proposition that is always true.
T T T TT F F TF T T TF F T T
p qpqpqp )(
pqp )(
Contradiction
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Contradiction - a compound proposition that is always false.
T F F
F T F
p pp p
Contingency
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A contingency is neither a tautology nor a contradiction.
T T T TT F F FF T F TF F F T
p qp q )( qpp )( qpp