Nsu module 01-logic-final

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Module #1 - Logic
North South University
Dept. of Computer Science & Engineering
CSE 173
Discrete Mathematics
Dr. Mohammad R. Rahman
Slides for a Course Based on the TextSlides for a Course Based on the Text
Discrete Mathematics & Its ApplicationsDiscrete Mathematics & Its Applications
(5(5thth
Edition)Edition)
by Kenneth H. Rosenby Kenneth H. Rosen

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Module #1:
The Fundamentals of Logic
Rosen 5Rosen 5thth
ed., §§1.1-1.4ed., §§1.1-1.4
~6-8 lectures~6-8 lectures

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Module #1 - Logic
Module #1: Foundations of Logic
(§§1.1-1.4)
Mathematical Logic is a tool for working with
elaborate compound statements. It includes:
• A formal language for expressing them.
• A concise notation for writing them.
• A methodology for objectively reasoning about
their truth or falsity.
• It is the foundation for expressing formal proofs in
all branches of mathematics.

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Module #1 - Logic
Foundations of Logic: Overview
• Propositional logic (Propositional logic (§1.1-1.2)§1.1-1.2)::
– Basic definitions. (§1.1)Basic definitions. (§1.1)
– Equivalence rules & derivations. (§1.2)Equivalence rules & derivations. (§1.2)
• Predicate logic (§1.3-1.4)Predicate logic (§1.3-1.4)
– Predicates.Predicates.
– Quantified predicate expressions.Quantified predicate expressions.
– Equivalences & derivations.Equivalences & derivations.

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Propositional Logic (§1.1)
Propositional LogicPropositional Logic is the logic of compoundis the logic of compound
statements built from simpler statementsstatements built from simpler statements
using so-calledusing so-called BooleanBoolean connectives.connectives.
Some applications in computer science:Some applications in computer science:
• Design of digital electronic circuits.Design of digital electronic circuits.
• Expressing conditions in programs.Expressing conditions in programs.
• Queries to databases & search engines.Queries to databases & search engines.
Topic #1 – Propositional Logic
George Boole
(1815-1864)
Chrysippus of Soli
(ca. 281 B.C. – 205 B.C.)

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Module #1 - Logic
Definition of a Proposition
Definition:Definition: AA propositionproposition (denoted(denoted pp,, qq,, rr, …) is simply:, …) is simply:
• aa statementstatement ((i.e.i.e., a declarative sentence), a declarative sentence)
– with some definite meaningwith some definite meaning, (not vague or ambiguous), (not vague or ambiguous)
• having ahaving a truth valuetruth value that’s eitherthat’s either truetrue ((TT) or) or falsefalse ((FF))
– it isit is nevernever both, neither, or somewhere “in between!”both, neither, or somewhere “in between!”
• However, you might notHowever, you might not knowknow the actual truth value,the actual truth value,
• and, the truth value mightand, the truth value might dependdepend on the situation or context.on the situation or context.
• However with probability theory,However with probability theory, we assignwe assign degrees of certaintydegrees of certainty
(“between”(“between” TT andand FF) to propositions.) to propositions.
– But for now: think True/False only!But for now: think True/False only!

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Examples of Propositions
• ““It is raining.”It is raining.” (In a given situation.)(In a given situation.)
• ““Beijing is the capital of China.” • “1 + 2 = 3”Beijing is the capital of China.” • “1 + 2 = 3”
But, the following areBut, the following are NOTNOT propositions:propositions:
• ““Who’s there?”Who’s there?” (interrogative, question)(interrogative, question)
• ““La la la la la.”La la la la la.” (meaningless interjection)(meaningless interjection)
• ““Just do it!”Just do it!” (imperative, command)(imperative, command)
• ““Yeah, I sorta dunno, whatever...”Yeah, I sorta dunno, whatever...” (vague)(vague)
• ““1 + 2”1 + 2” (expression with a non-true/false value)(expression with a non-true/false value)

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AnAn operatoroperator oror connectiveconnective combines one orcombines one or
moremore operandoperand expressions into a largerexpressions into a larger
expression. (expression. (E.g.E.g., “+” in numeric exprs.), “+” in numeric exprs.)
• UnaryUnary operators take 1 operand (operators take 1 operand (e.g.,e.g., −−3);3);
binarybinary operators take 2 operands (operators take 2 operands (egeg 33 ×× 4).4).
• PropositionalPropositional oror BooleanBoolean operatorsoperators operateoperate
on propositionson propositions (or their truth values)(or their truth values)
instead of on numbers.instead of on numbers.
Operators / Connectives
Topic #1.0 – Propositional Logic: Operators

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Module #1 - Logic
Some Popular Boolean Operators
Formal NameFormal Name NicknameNickname ArityArity SymbolSymbol
Negation operatorNegation operator NOTNOT UnaryUnary ¬¬
Conjunction operatorConjunction operator ANDAND BinaryBinary ∧∧
Disjunction operatorDisjunction operator OROR BinaryBinary ∨∨
Exclusive-OR operatorExclusive-OR operator XORXOR BinaryBinary ⊕⊕
Implication operatorImplication operator IMPLIESIMPLIES BinaryBinary →→
Biconditional operatorBiconditional operator IFFIFF BinaryBinary ↔↔

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The Negation Operator
The unaryThe unary negation operatornegation operator “¬” (“¬” (NOTNOT))
transforms a prop. into its logicaltransforms a prop. into its logical negationnegation..
E.g.E.g. IfIf pp = “I have brown hair.”= “I have brown hair.”
then ¬then ¬pp = “I do= “I do notnot have brown hair.”have brown hair.”
TheThe truth tabletruth table for NOT:for NOT: p ¬p
T F
F T
T :≡ True; F :≡ False
“:≡” means “is defined as”
Operand
column
Result
column

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Module #1 - Logic
The Conjunction Operator
The binaryThe binary conjunction operatorconjunction operator ““∧∧” (” (ANDAND))
combines two propositions to form theircombines two propositions to form their
logicallogical conjunctionconjunction..
E.g.E.g. IfIf pp=“I will have salad for lunch.” and=“I will have salad for lunch.” and
q=q=“I will have steak for dinner.”, then“I will have steak for dinner.”, then
pp∧∧qq=“I will have salad for lunch=“I will have salad for lunch andand
I will have steak for dinner.”I will have steak for dinner.”
Remember: “∧∧” points up like an “A”, and it means “” points up like an “A”, and it means “∧∧NDND””
∧∧NDND

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• Note that aNote that a
conjunctionconjunction
pp11 ∧∧ pp22 ∧∧ …… ∧∧ ppnn
ofof nn propositionspropositions
will have 2will have 2nn
rowsrows
in its truth table.in its truth table.
• Also: ¬ andAlso: ¬ and ∧∧ operations together are suffi-operations together are suffi-
cient to expresscient to express anyany Boolean truth table!Boolean truth table!
Conjunction Truth Table
p q p∧q
F F F
F T F
T F F
T T T
Operand columns

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The Disjunction Operator
The binaryThe binary disjunction operatordisjunction operator ““∨∨” (” (OROR))
logicallogical disjunctiondisjunction..
pp=“My car has a bad engine.”=“My car has a bad engine.”
q=q=“My car has a bad carburetor.”“My car has a bad carburetor.”
pp∨∨qq=“Either my car has a bad engine,=“Either my car has a bad engine, oror
my car has a bad carburetor.”my car has a bad carburetor.” After the downward-
pointing “axe” of “∨∨””
splits the wood, yousplits the wood, you
can take 1 piece OR thecan take 1 piece OR the
other, or both.other, or both.
∨∨
Meaning is like “and/or” in English.

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Module #1 - Logic
• Note thatNote that pp∨∨qq meansmeans
thatthat pp is true, oris true, or qq isis
true,true, or bothor both are true!are true!
• So, this operation isSo, this operation is
also calledalso called inclusive or,inclusive or,
because itbecause it includesincludes thethe
possibility that bothpossibility that both pp andand qq are true.are true.
• ““¬” and “¬” and “∨∨” together are also universal.” together are also universal.
Disjunction Truth Table
p q p∨ q
F F F
F T T
T F T
T T T
Note
difference
from AND

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Nested Propositional Expressions
• Use parentheses toUse parentheses to group sub-expressionsgroup sub-expressions::
““I just saw my oldI just saw my old ffriendriend, and either, and either he’she’s
ggrownrown oror I’veI’ve sshrunkhrunk.” =.” = ff ∧∧ ((gg ∨∨ ss))
– ((ff ∧∧ gg)) ∨∨ ss would mean something differentwould mean something different
– ff ∧∧ gg ∨∨ ss would be ambiguouswould be ambiguous
• By convention, “¬” takesBy convention, “¬” takes precedenceprecedence overover
both “both “∧∧” and “” and “∨∨”.”.
– ¬¬ss ∧∧ ff means (¬means (¬ss)) ∧∧ ff ,, notnot ¬ (¬ (ss ∧∧ ff))

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Module #1 - Logic
A Simple Exercise
LetLet pp=“It rained last night”,=“It rained last night”,
qq=“The sprinklers came on last night,”=“The sprinklers came on last night,”
rr=“The lawn was wet this morning.”=“The lawn was wet this morning.”
Translate each of the following into English:Translate each of the following into English:
¬¬pp ==
rr ∧∧ ¬¬pp ==
¬¬ rr ∨∨ pp ∨∨ q =q =
“It didn’t rain last night.”
“The lawn was wet this morning, and
it didn’t rain last night.”
“Either the lawn wasn’t wet this
morning, or it rained last night, or
the sprinklers came on last night.”

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The Exclusive Or Operator
The binaryThe binary exclusive-or operatorexclusive-or operator ““⊕⊕” (” (XORXOR))
logical “exclusive or” (exjunction?).logical “exclusive or” (exjunction?).
pp = “I will earn an A in this course,”= “I will earn an A in this course,”
qq == “I will drop this course,”“I will drop this course,”
pp ⊕⊕ qq = “I will either earn an A in this course,= “I will either earn an A in this course,
or I will drop it (but not both!)”or I will drop it (but not both!)”

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Module #1 - Logic
• Note thatNote that pp⊕⊕qq meansmeans
thatthat pp is true, oris true, or qq isis
true, buttrue, but not bothnot both!!
• This operation isThis operation is
calledcalled exclusive or,exclusive or,
because itbecause it excludesexcludes thethe
possibility that bothpossibility that both pp andand qq are true.are true.
• ““¬” and “¬” and “⊕⊕” together are” together are notnot universal.universal.
Exclusive-Or Truth Table
p q p⊕ q
F F F
F T T
T F T
T T F Note
difference
from OR.

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Module #1 - Logic
Note thatNote that EnglishEnglish “or” can be“or” can be ambiguousambiguous
regarding the “both” case!regarding the “both” case!
““Pat is a singer orPat is a singer or
Pat is a writer.” -Pat is a writer.” -
““Pat is a man orPat is a man or
Pat is a woman.” -Pat is a woman.” -
Need context to disambiguate the meaning!Need context to disambiguate the meaning!
For this class, assume “or” meansFor this class, assume “or” means inclusiveinclusive..
Natural Language is Ambiguous
p q p "or" q
F F F
F T T
T F T
T T ?
∨
⊕

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Module #1 - Logic
The Implication Operator
TheThe implicationimplication pp →→ qq states thatstates that pp impliesimplies q.q.
I.e.I.e., If, If pp is true, thenis true, then qq is true; but ifis true; but if pp is notis not
true, thentrue, then qq could be either true or false.could be either true or false.
E.g.E.g., let, let pp = “You study hard.”= “You study hard.”
qq = “You will get a good grade.”= “You will get a good grade.”
pp →→ q =q = “If you study hard, then you will get“If you study hard, then you will get
a good grade.”a good grade.” (else, it could go either way)(else, it could go either way)
antecedent consequent

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Implication Truth Table
• pp →→ qq isis falsefalse onlyonly whenwhen
pp is true butis true but qq isis notnot true.true.
• pp →→ qq doesdoes notnot saysay
thatthat pp causescauses qq!!
• pp →→ qq doesdoes notnot requirerequire
thatthat pp oror qq are ever trueare ever true!!
• E.g.E.g. “(1=0)“(1=0) →→ pigs can fly” is TRUE!pigs can fly” is TRUE!
p q p→q
F F T
F T T
T F F
T T T
The
only
False
case!

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Module #1 - Logic
Examples of Implications
• ““If this lecture ever ends, then the sun willIf this lecture ever ends, then the sun will
rise tomorrow.”rise tomorrow.” TrueTrue oror FalseFalse??
• ““If Tuesday is a day of the week, then I amIf Tuesday is a day of the week, then I am
a penguin.”a penguin.” TrueTrue oror FalseFalse??
• ““If 1+1=6, then Bush is president.”If 1+1=6, then Bush is president.”
TrueTrue oror FalseFalse??
• ““If the moon is made of green cheese, then IIf the moon is made of green cheese, then I
am richer than Bill Gates.”am richer than Bill Gates.” TrueTrue oror FalseFalse??

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Difference between English
Language and Logic
• The way we have defined implications is more generalThe way we have defined implications is more general
than the meaning attached to English Language.than the meaning attached to English Language.
• Consider a sentence like,Consider a sentence like,
– ““If it is sunny today, then we will go to beach”If it is sunny today, then we will go to beach”
• In normal language there is a relationship betweenIn normal language there is a relationship between
antecedent and consequent.antecedent and consequent.
• Now Let us consider this exampleNow Let us consider this example
– ““If it is Friday, then 2+3=5”If it is Friday, then 2+3=5”
• In logic, we consider the sentenceIn logic, we consider the sentence TrueTrue since thesince the
conclusion is trueconclusion is true

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Module #1 - Logic
More about Discrepancy
• In English, a sentence “ifIn English, a sentence “if pp thenthen qq” usually really” usually really
implicitlyimplicitly means something like,means something like,
• ““In all possible situationsIn all possible situations, if, if pp thenthen qq.”.”
– That is, “ForThat is, “For pp to be true andto be true and qq false isfalse is impossibleimpossible.”.”
– Or, “IOr, “I guaranteeguarantee that no matter what, ifthat no matter what, if pp, then, then qq.”.”
• In mathematical concept of an implication isIn mathematical concept of an implication is
independent of a cause and effect relationshipindependent of a cause and effect relationship
between hypothesis and conclusionbetween hypothesis and conclusion
• The implication only specifies the truth values; it isThe implication only specifies the truth values; it is
not based on English usage.not based on English usage.

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Module #1 - Logic
English Phrases Meaning p → q
• ““pp impliesimplies qq””
• ““ifif pp, then, then qq””
• ““ifif pp,, qq””
• ““whenwhen pp,, qq””
• ““wheneverwhenever pp,, qq””
• ““qq ifif pp””
• ““qq whenwhen pp””
• ““qq wheneverwhenever pp””
• ““pp only ifonly if qq””
• ““pp is sufficient foris sufficient for qq””
• ““qq is necessary foris necessary for pp””
• ““qq follows fromfollows from pp””
• ““qq is implied byis implied by pp””
We will see some equivalentWe will see some equivalent
logic expressions later.logic expressions later.

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Module #1 - Logic
Converse, Inverse, Contrapositive
Some terminology, for an implicationSome terminology, for an implication pp →→ qq::
• ItsIts converseconverse is:is: qq →→ pp..
• ItsIts inverseinverse is:is: ¬¬pp →→ ¬¬qq..
• ItsIts contrapositivecontrapositive:: ¬¬qq →→ ¬¬ p.p.
• One of these three has theOne of these three has the same meaningsame meaning
(same truth table) as(same truth table) as pp →→ qq. Can you figure. Can you figure
out which?out which?

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How do we know for sure?
Proving the equivalence ofProving the equivalence of pp →→ qq and itsand its
contrapositive using truth tables:contrapositive using truth tables:
p q ¬q ¬p p→q ¬q →¬p
F F T T T T
F T F T T T
T F T F F F
T T F F T T

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Module #1 - Logic
The biconditional operator
• TheThe biconditionalbiconditional pp ↔ qq is theis the
proposition that is true when p andproposition that is true when p and
q have the same truth values, and isq have the same truth values, and is
false otherwise.false otherwise.
• pp ↔ qq is true preciously when bothis true preciously when both
the implicationsthe implications
pp →→ qq andand qq →→ pp are trueare true
• The terminology “if and only if q”The terminology “if and only if q”
is used foris used for pp ↔ qq
p q p ↔q
F F T
F T F
T F F
T T T

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Module #1 - Logic
Biconditional Truth Table
• pp ↔ qq means thatmeans that pp andand qq
have thehave the samesame truth value.truth value.
• Note this truth table is theNote this truth table is the
exactexact oppositeopposite ofof ⊕⊕’s!’s!
Thus,Thus, pp ↔ qq means ¬(means ¬(pp ⊕⊕ qq))
• pp ↔ qq doesdoes notnot implyimply
thatthat pp andand qq are true, or that either of them causesare true, or that either of them causes
the other, or that they have a common cause.the other, or that they have a common cause.
p q p ↔q
F F T
F T F
T F F
T T T

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English Equivalent
• ““p is necessary and sufficient for q”p is necessary and sufficient for q”
• ““if p then q, and conversely.”if p then q, and conversely.”
• ““p iff q”p iff q”
• Example: p: You can take the flight and q:Example: p: You can take the flight and q:
You buy a ticketYou buy a ticket
• pp ↔ q: “ You can take the flight if and onlyq: “ You can take the flight if and only
if you buy a ticket”if you buy a ticket”

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Translating English Sentences
1.1. You can access the Internet from campusYou can access the Internet from campus
only if you are a computer science majoronly if you are a computer science major
or you are not a freshman.or you are not a freshman.
2.2. You cannot ride the roller coaster if youYou cannot ride the roller coaster if you
are under 4 feet tall unless you are olderare under 4 feet tall unless you are older
than 16 years old.than 16 years old.
3.3. The automated reply cannot be sent whenThe automated reply cannot be sent when
the file system is full.the file system is full.

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System Specifications
• System and software engineers take requirements inSystem and software engineers take requirements in
natural language and produce precise and unambiguousnatural language and produce precise and unambiguous
specifications that could be used as the basis of systemspecifications that could be used as the basis of system
development.development.
• System specifications should not contain conflictingSystem specifications should not contain conflicting
requirements.requirements.
• Propositional expressions representing these specificationsPropositional expressions representing these specifications
need to beneed to be consistentconsistent ::
• there must be an assignment of truth values to the variablethere must be an assignment of truth values to the variable
in the expressions that makes all the expressions truein the expressions that makes all the expressions true

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Module #1 - Logic
Example: System Specifications
Are the system specifications consistent?Are the system specifications consistent?
• The diagnostic message is stored in theThe diagnostic message is stored in the
buffer or it is retransmittedbuffer or it is retransmitted
• The diagnostic message is not stored in theThe diagnostic message is not stored in the
buffer.buffer.
• If the diagnostic message is stored in theIf the diagnostic message is stored in the
buffer, then it is retransmitted.buffer, then it is retransmitted.

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Some Alternative Notations
Name: not and or xor implies iff
Propositional logic: ¬∧ ∨ ⊕ → ↔
Boolean algebra: p pq + ⊕
C/C++/Java (wordwise): ! && || != ==
C/C++/Java (bitwise): ~ & | ^
Logic gates:

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Module #1 - Logic
Bits and Bit Operations
• AA bitbit is ais a bibinary (base 2) dignary (base 2) digitit: 0 or 1.: 0 or 1.
• Bits may be used to represent truth values.Bits may be used to represent truth values.
• By convention:By convention:
0 represents “false”; 1 represents “true”.0 represents “false”; 1 represents “true”.
• Boolean algebraBoolean algebra is like ordinary algebrais like ordinary algebra
except that variables stand for bits, + meansexcept that variables stand for bits, + means
“or”, and multiplication means “and”.“or”, and multiplication means “and”.
Topic #2 – Bits
John Tukey
(1915-2000)

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Bitwise Operations
• Boolean operations can be extended toBoolean operations can be extended to
operate on bit strings as well as single bits.operate on bit strings as well as single bits.
• E.g.:E.g.:
01 1011 011001 1011 0110
11 0001 110111 0001 1101
11 1011 1111 Bit-wise ORBit-wise OR
01 0001 0100 Bit-wise ANDBit-wise AND
10 1010 1011 Bit-wise XORBit-wise XOR
Topic #2 – Bits

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End of §1.1
You have learned about:You have learned about:
• Propositions: WhatPropositions: What
they are.they are.
• Propositional logicPropositional logic
operators’operators’
– Symbolic notations.Symbolic notations.
– English equivalents.English equivalents.
– Logical meaning.Logical meaning.
– Truth tables.Truth tables.
• Atomic vs. compoundAtomic vs. compound
propositions.propositions.
• Alternative notations.Alternative notations.
• Bits and bit-strings.Bits and bit-strings.
• Next section: §1.2Next section: §1.2
– PropositionalPropositional
equivalences.equivalences.
– How to prove them.How to prove them.

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Propositional Equivalence (§1.2)
TwoTwo syntacticallysyntactically ((i.e.,i.e., textually) differenttextually) different
compound propositions may be thecompound propositions may be the
semanticallysemantically identical (identical (i.e.,i.e., have the samehave the same
meaning). We call themmeaning). We call them equivalentequivalent. Learn:. Learn:
• VariousVarious equivalence rulesequivalence rules oror lawslaws..
• How toHow to proveprove equivalences usingequivalences using symbolicsymbolic
derivationsderivations..
Topic #1.1 – Propositional Logic: Equivalences

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Module #1 - Logic
Tautologies and Contradictions
AA tautologytautology is a compound proposition that isis a compound proposition that is
truetrue no matter whatno matter what the truth values of itsthe truth values of its
atomic propositions are!atomic propositions are!
Ex.Ex. pp ∨∨ ¬¬pp [What is its truth table?][What is its truth table?]
AA contradictioncontradiction is a compound propositionis a compound proposition
that isthat is falsefalse no matter what!no matter what! Ex.Ex. pp ∧∧ ¬¬pp
[Truth table?][Truth table?]
Other compound props. areOther compound props. are contingenciescontingencies..

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Module #1 - Logic
Logical Equivalence
Compound propositionCompound proposition pp isis logicallylogically
equivalentequivalent to compound propositionto compound proposition qq,,
writtenwritten pp⇔⇔qq,, IFFIFF the compoundthe compound
propositionproposition pp↔↔qq is a tautology.is a tautology.
Compound propositionsCompound propositions pp andand qq are logicallyare logically
equivalent to each otherequivalent to each other IFFIFF pp andand qq
contain the same truth values as each othercontain the same truth values as each other
inin allall rows of their truth tables.rows of their truth tables.

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Ex.Ex. Prove thatProve that pp∨∨qq ⇔⇔ ¬¬((¬¬pp ∧∧ ¬¬qq).).
p q pp∨∨qq ¬¬pp ¬¬qq ¬¬pp ∧∧¬¬qq ¬¬((¬¬pp ∧∧¬¬qq))
F F
F T
T F
T T
Proving Equivalence
via Truth Tables
F
T
T
T
T
T
T
T
T
T
F
F
F
F
F
F
F
F
T
T

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Module #1 - Logic
Equivalence Laws - Examples
• IdentityIdentity:: pp∧∧TT ⇔⇔ p pp p∨∨FF ⇔⇔ pp
• DominationDomination:: pp∨∨TT ⇔⇔ TT pp∧∧FF ⇔⇔ FF
• IdempotentIdempotent:: pp∨∨pp ⇔⇔ p pp p∧∧pp ⇔⇔ pp
• Double negation:Double negation: ¬¬¬¬pp ⇔⇔ pp
• Commutative:Commutative: pp∨∨qq ⇔⇔ qq∨∨p pp p∧∧qq ⇔⇔ qq∧∧pp
• Associative:Associative: ((pp∨∨qq))∨∨rr ⇔⇔ pp∨∨((qq∨∨rr))
((pp∧∧qq))∧∧rr ⇔⇔ pp∧∧((qq∧∧rr))

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More Equivalence Laws
• DistributiveDistributive:: pp∨∨((qq∧∧rr)) ⇔⇔ ((pp∨∨qq))∧∧((pp∨∨rr))
pp∧∧((qq∨∨rr)) ⇔⇔ ((pp∧∧qq))∨∨((pp∧∧rr))
• De Morgan’sDe Morgan’s::
¬¬((pp∧∧qq)) ⇔⇔ ¬¬pp ∨∨ ¬¬qq
¬¬((pp∨∨qq)) ⇔⇔ ¬¬pp ∧∧ ¬¬qq
• Trivial tautology/contradictionTrivial tautology/contradiction::
pp ∨∨ ¬¬pp ⇔⇔ TT pp ∧∧ ¬¬pp ⇔⇔ FF
Augustus
De Morgan
(1806-1871)

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Module #1 - Logic
Defining Operators via Equivalences
Using equivalences, we canUsing equivalences, we can definedefine operatorsoperators
in terms of other operators.in terms of other operators.
• Exclusive or:Exclusive or: pp⊕⊕qq ⇔⇔ ((pp∨∨qq))∧¬∧¬((pp∧∧qq))
pp⊕⊕qq ⇔⇔ ((pp∧¬∧¬qq))∨∨((qq∧¬∧¬pp))
• Implies:Implies: pp→→qq ⇔⇔ ¬¬pp ∨∨ qq
• Biconditional:Biconditional: pp↔↔qq ⇔⇔ ((pp→→qq)) ∧∧ ((qq→→pp))
pp↔↔qq ⇔⇔ ¬¬((pp⊕⊕qq))

03/07/17 45
Module #1 - Logic
An Example Problem
• Check using a symbolic derivation whetherCheck using a symbolic derivation whether
((pp ∧∧ ¬¬qq)) →→ ((pp ⊕⊕ rr)) ⇔⇔ ¬¬pp ∨∨ qq ∨∨ ¬¬rr..
((pp ∧∧ ¬¬qq)) →→ ((pp ⊕⊕ rr)) [Expand definition of[Expand definition of →→]]
⇔⇔ ¬¬((pp ∧∧ ¬¬qq)) ∨∨ ((pp ⊕⊕ rr)) [Expand defn. of[Expand defn. of ⊕⊕]]
⇔⇔ ¬¬((pp ∧∧ ¬¬qq)) ∨∨ ((((pp ∨∨ rr)) ∧∧ ¬¬((pp ∧∧ rr))))
[DeMorgan’s Law][DeMorgan’s Law]
⇔⇔ ((¬¬pp ∨∨ qq)) ∨∨ ((((pp ∨∨ rr)) ∧∧ ¬¬((pp ∧∧ rr))))
cont.cont.

03/07/17 46
Module #1 - Logic
Example Continued...
((¬¬pp ∨∨ qq)) ∨∨ ((((pp ∨∨ rr)) ∧∧ ¬¬((pp ∧∧ rr)))) ⇔⇔ [[∨∨ commutes]commutes]
⇔⇔ ((qq ∨∨ ¬¬pp)) ∨∨ ((((pp ∨∨ rr)) ∧∧ ¬¬((pp ∧∧ rr)))) [[∨∨ associative]associative]
⇔⇔ qq ∨∨ ((¬¬pp ∨∨ ((((pp ∨∨ rr)) ∧∧ ¬¬((pp ∧∧ rr))))) [distrib.) [distrib. ∨∨ overover ∧∧]]
⇔⇔ qq ∨∨ ((((((¬¬pp ∨∨ ((pp ∨∨ rr)))) ∧∧ ((¬¬pp ∨∨ ¬¬((pp ∧∧ rr))))))
[assoc.][assoc.] ⇔⇔ qq ∨∨ ((((((¬¬pp ∨∨ pp)) ∨∨ rr)) ∧∧ ((¬¬pp ∨∨ ¬¬((pp ∧∧ rr))))))
[trivail taut.][trivail taut.] ⇔⇔ qq ∨∨ ((((TT ∨∨ rr)) ∧∧ ((¬¬pp ∨∨ ¬¬((pp ∧∧ rr))))))
[domination][domination] ⇔⇔ qq ∨∨ ((TT ∧∧ ((¬¬pp ∨∨ ¬¬((pp ∧∧ rr))))))
[identity][identity] ⇔⇔ qq ∨∨ ((¬¬pp ∨∨ ¬¬((pp ∧∧ rr)))) ⇔⇔ cont.cont.

03/07/17 47
Module #1 - Logic
End of Long Example
qq ∨∨ ((¬¬pp ∨∨ ¬¬((pp ∧∧ rr))))
[DeMorgan’s][DeMorgan’s] ⇔⇔ qq ∨∨ ((¬¬pp ∨∨ ((¬¬pp ∨∨ ¬¬rr))))
[Assoc.][Assoc.] ⇔⇔ qq ∨∨ ((((¬¬pp ∨∨ ¬¬pp)) ∨∨ ¬¬rr))
[Idempotent][Idempotent] ⇔⇔ qq ∨∨ ((¬¬pp ∨∨ ¬¬rr))
[Assoc.][Assoc.] ⇔⇔ ((qq ∨∨ ¬¬pp)) ∨∨ ¬¬rr
[Commut.][Commut.] ⇔⇔ ¬¬pp ∨∨ qq ∨∨ ¬¬rr

03/07/17 48
Module #1 - Logic
Review: Propositional Logic
(§§1.1-1.2)
• Atomic propositions:Atomic propositions: pp,, qq,, rr, …, …
• Boolean operators:Boolean operators: ¬¬ ∧∧ ∨∨ ⊕⊕ →→ ↔↔
• Compound propositions: s :≡ (p ∧∧ ¬¬qq)) ∨∨ rr
• Equivalences:Equivalences: pp∧¬∧¬qq ⇔⇔ ¬¬((pp →→ qq))
• Proving equivalences using:Proving equivalences using:
– Truth tables.Truth tables.
– Symbolic derivations.Symbolic derivations. pp ⇔⇔ qq ⇔⇔ r …r …

03/07/17 49
Module #1 - Logic
Predicate Logic (§1.3)
• Predicate logicPredicate logic is an extension ofis an extension of
propositional logic that permits conciselypropositional logic that permits concisely
reasoning about wholereasoning about whole classesclasses of entities.of entities.
• Propositional logic (recall) treats simplePropositional logic (recall) treats simple
propositionspropositions (sentences) as atomic entities.(sentences) as atomic entities.
• In contrast,In contrast, predicatepredicate logic distinguishes thelogic distinguishes the
subjectsubject of a sentence from itsof a sentence from its predicate.predicate.
– Remember these English grammar terms?Remember these English grammar terms?
Topic #3 – Predicate Logic

03/07/17 50
Module #1 - Logic
Subjects and Predicates
• In the sentence “The dog is sleeping”:In the sentence “The dog is sleeping”:
– The phrase “the dog” denotes theThe phrase “the dog” denotes the subjectsubject --
thethe objectobject oror entityentity that the sentence is about.that the sentence is about.
– The phrase “is sleeping” denotes theThe phrase “is sleeping” denotes the predicatepredicate--
a property that is truea property that is true ofof the subject.the subject.
• In predicate logic, aIn predicate logic, a predicatepredicate is modeled asis modeled as
aa functionfunction PP(·) from objects to propositions.(·) from objects to propositions.
– PP((xx) = “) = “xx is sleeping” (whereis sleeping” (where xx is any object).is any object).

03/07/17 51
Module #1 - Logic
More About Predicates
• Convention: Lowercase variablesConvention: Lowercase variables xx,, yy,, z...z... denotedenote
objects/entities; uppercase variablesobjects/entities; uppercase variables PP,, QQ,, RR……
denote propositional functions (predicates).denote propositional functions (predicates).
• Keep in mind that theKeep in mind that the result ofresult of applyingapplying aa
predicatepredicate PP to an objectto an object xx is theis the proposition Pproposition P((xx).).
But the predicateBut the predicate PP itselfitself ((e.g. Pe.g. P=“is sleeping”) is=“is sleeping”) is
notnot a proposition (not a complete sentence).a proposition (not a complete sentence).
– E.g.E.g. ifif PP((xx) = “) = “xx is a prime number”,is a prime number”,
PP(3) is the(3) is the propositionproposition “3 is a prime number.”“3 is a prime number.”

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Module #1 - Logic
Propositional Functions
• Predicate logicPredicate logic generalizesgeneralizes the grammaticalthe grammatical
notion of a predicate to also includenotion of a predicate to also include
propositional functions ofpropositional functions of anyany number ofnumber of
arguments, each of which may takearguments, each of which may take anyany
grammatical role that a noun can take.grammatical role that a noun can take.
– E.g.E.g. letlet PP((xx,,y,zy,z) = “) = “xx gavegave yy the gradethe grade zz”, then if”, then if
x=x=“Mike”,“Mike”, yy=“Mary”,=“Mary”, zz=“A”, then=“A”, then PP((xx,,yy,,zz) =) =
“Mike gave Mary the grade A.”“Mike gave Mary the grade A.”

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Module #1 - Logic
Universes of Discourse (U.D.s)
• The power of distinguishing objects fromThe power of distinguishing objects from
predicates is that it lets you state thingspredicates is that it lets you state things
aboutabout manymany objects at once.objects at once.
• E.g., letE.g., let PP((xx)=“)=“xx+1>+1>xx”. We can then say,”. We can then say,
“For“For anyany numbernumber xx,, PP((xx) is true” instead of) is true” instead of
((00+1>+1>00)) ∧∧ ((11+1>+1>11)) ∧∧ ((22+1>+1>22)) ∧∧ ......
• The collection of values that a variableThe collection of values that a variable xx
can take is calledcan take is called xx’s’s universe of discourseuniverse of discourse..

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Module #1 - Logic
Quantifier Expressions
• QuantifiersQuantifiers provide a notation that allowsprovide a notation that allows
us tous to quantifyquantify (count)(count) how manyhow many objects inobjects in
the univ. of disc. satisfy a given predicate.the univ. of disc. satisfy a given predicate.
• ““∀∀”” is the FORis the FOR∀∀LL orLL or universaluniversal quantifier.quantifier.
∀∀xx PP((xx) means) means for allfor all x in the u.d.,x in the u.d., PP holds.holds.
• ““∃∃”” is theis the ∃∃XISTS orXISTS or existentialexistential quantifier.quantifier.
∃∃x Px P((xx) means) means therethere existsexists anan xx in the u.d.in the u.d.
(that is, 1 or more)(that is, 1 or more) such thatsuch that PP((xx) is true.) is true.

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Module #1 - Logic
The Universal Quantifier ∀
• Example:Example:
Let the u.d. of x beLet the u.d. of x be parking spaces at NSUparking spaces at NSU..
LetLet PP((xx) be the) be the predicatepredicate ““xx is full.”is full.”
Then theThen the universal quantification of Puniversal quantification of P((xx),),
∀∀xx PP((xx), is the), is the proposition:proposition:
– ““All parking spaces at NSU are full.”All parking spaces at NSU are full.”
– i.e.i.e., “Every parking space at NSU is full.”, “Every parking space at NSU is full.”
– i.e.i.e., “For each parking space at NSU, that space is full.”, “For each parking space at NSU, that space is full.”

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Module #1 - Logic
The Existential Quantifier ∃
• Example:Example:
Let the u.d. of x beLet the u.d. of x be parking spaces at NSUparking spaces at NSU..
LetLet PP((xx) be the) be the predicatepredicate ““xx is full.”is full.”
Then theThen the existential quantification of Pexistential quantification of P((xx),),
∃∃xx PP((xx), is the), is the propositionproposition::
– ““Some parking space at NSU is full.”Some parking space at NSU is full.”
– ““There is a parking space at NSU that is full.”There is a parking space at NSU that is full.”
– ““At least one parking space at NSU is full.”At least one parking space at NSU is full.”

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Module #1 - Logic
Quantifier Equivalence Laws
• Definitions of quantifiers: If u.d.=a,b,c,…Definitions of quantifiers: If u.d.=a,b,c,…
∀∀x Px P((xx)) ⇔⇔ PP(a)(a) ∧∧ PP(b)(b) ∧∧ PP(c)(c) ∧∧ ……
∃∃x Px P((xx)) ⇔⇔ PP(a)(a) ∨∨ PP(b)(b) ∨∨ PP(c)(c) ∨∨ ……
• From those, we can prove the laws:From those, we can prove the laws:
¬∀¬∀x Px P((xx)) ⇔⇔ ∃∃xx ¬¬PP((xx))
¬∃¬∃x Px P((xx)) ⇔⇔ ∀∀xx ¬¬PP((xx))
• WhichWhich propositionalpropositional equivalence laws canequivalence laws can
be used to prove this?be used to prove this?

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Module #1 - Logic
Free and Bound Variables
• An expression likeAn expression like PP((xx) is said to have a) is said to have a
free variablefree variable xx (meaning,(meaning, xx is undefined).is undefined).
• A quantifier (eitherA quantifier (either ∀∀ oror ∃∃)) operatesoperates on anon an
expression having one or more freeexpression having one or more free
variables, andvariables, and bindsbinds one or more of thoseone or more of those
variables, to produce an expression havingvariables, to produce an expression having
one or moreone or more boundbound variablesvariables..

03/07/17 59
Module #1 - Logic
Example of Binding
• PP((x,yx,y) has 2 free variables,) has 2 free variables, xx andand yy..
∀ ∀∀xx PP((xx,,yy) has 1 free variable, and one bound) has 1 free variable, and one bound
variable.variable. [Which is which?][Which is which?]
• ““PP((xx), where), where xx=3” is another way to bind=3” is another way to bind xx..
• An expression withAn expression with zerozero free variables is a bona-free variables is a bona-
fide (actual) proposition.fide (actual) proposition.
• An expression withAn expression with one or moreone or more free variables isfree variables is
still only a predicate:still only a predicate: e.g.e.g. letlet QQ((yy) =) = ∀∀xx PP((xx,,yy))

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Module #1 - Logic
Nesting of Quantifiers
Example: Let the u.d. ofExample: Let the u.d. of xx && yy be people.be people.
LetLet LL((xx,,yy)=“)=“xx likeslikes yy”” (a predicate w. 2 f.v.’s)(a predicate w. 2 f.v.’s)
ThenThen ∃∃y Ly L((x,yx,y) = “There is someone whom) = “There is someone whom xx
likes.”likes.” (A predicate w. 1 free variable,(A predicate w. 1 free variable, xx))
ThenThen ∀∀xx ((∃∃y Ly L((x,yx,y)) =)) =
“Everyone has someone whom they like.”“Everyone has someone whom they like.”
(A __________ with ___ free variables.)(A __________ with ___ free variables.)

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Module #1 - Logic
Quantifier Exercise
IfIf RR((xx,,yy)=“)=“xx relies uponrelies upon yy,” express the,” express the
following in unambiguous English:following in unambiguous English:
∀∀xx((∃∃y Ry R((x,yx,y))=))=
∃∃yy((∀∀xx RR((x,yx,y))=))=
∃∃xx((∀∀y Ry R((x,yx,y))=))=
∀∀yy((∃∃x Rx R((x,yx,y))=))=
∀∀xx((∀∀y Ry R((x,yx,y))=))=
Everyone has someone to rely on.
There’s a poor overburdened soul whom
everyone relies upon (including himself)!
There’s some needy person who relies
upon everybody (including himself).
Everyone has someone who relies upon them.
Everyone relies upon everybody,
(including themselves)!

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Module #1 - Logic
Nested quantifiers
AssumeAssume S(x,y)S(x,y) means “means “xx seessees yy””
u.d.=all peopleu.d.=all people
What does the following formula mean?What does the following formula mean?
∀∀xSxS((x,ax,a))

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Module #1 - Logic
Nested quantifiers
AssumeAssume S(x,y)S(x,y) means “means “xx seessees yy”.”.
u.d.=all peopleu.d.=all people
∀∀xSxS((x,ax,a) means) means
“For every“For every xx,, xx seessees aa””
In other words,In other words,
“Everyone sees“Everyone sees aa””

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Module #1 - Logic
Nested quantifiers
What does the following formula mean?What does the following formula mean?
∀∀xx((∃∃y Sy S((x,yx,y))))

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Module #1 - Logic
Nested quantifiers
∀∀xx((∃∃y Sy S((x,yx,y)) means “)) means “For everyFor every xx, there exists, there exists
aa yy such thatsuch that xx seessees y”y”
In other words:In other words:
“Everyone sees someone”“Everyone sees someone”

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Module #1 - Logic
Interactions between
quantifiers and connectives
Let the u.d. beLet the u.d. be parking spaces at NSUparking spaces at NSU..
LetLet PP((xx) be “) be “xx is occupied.”is occupied.”
Let Q(Let Q(xx) be “) be “xx is free of charge.”is free of charge.”
1.1. ∃∃xx ((QQ((xx)) ∧∧ PP((xx))))
2.2. ∀∀xx ((QQ((xx)) ∧∧ PP((xx))))
3.3. ∀∀xx ((QQ((xx)) →→PP((xx))))
4.4. ∃∃xx ((QQ((xx)) →→ PP((xx))))

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Module #1 - Logic
I. Construct English paraphrases
Let the u.d. beLet the u.d. be parking spaces at NSUparking spaces at NSU..
LetLet PP((xx) be “) be “xx is occupied.”is occupied.”
Let Q(Let Q(xx) be “) be “xx is free of charge.”is free of charge.”
1.1. ∃∃xx ((QQ((xx)) ∧∧ PP((xx))))
2.2. ∀∀xx ((QQ((xx)) ∧∧ PP((xx))))
3.3. ∀∀xx ((QQ((xx)) →→PP((xx))))
4.4. ∃∃xx ((QQ((xx)) →→ PP((xx))))

03/07/17 68
Module #1 - Logic
I. Construct English paraphrases
1.1. ∃∃xx ((QQ((xx)) ∧∧ PP((xx))) Some places are free of) Some places are free of
charge and occupiedcharge and occupied
2.2. ∀∀xx ((QQ((xx)) ∧∧ PP((xx))) All places are free of) All places are free of
charge and occupiedcharge and occupied
3.3. ∀∀xx ((QQ((xx)) →→PP((xx))) All places that are free) All places that are free
of charge are occupiedof charge are occupied
4.4. ∃∃xx ((QQ((xx)) →→ PP((xx))) For some places x, if x) For some places x, if x
is free of charge then x is occupiedis free of charge then x is occupied

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Module #1 - Logic
About the last of these
4.4. ∃∃xx ((QQ((xx)) →→ PP((xx))) “For some x, if x is free of) “For some x, if x is free of
charge then x is occupied”charge then x is occupied”
∃∃xx ((QQ((xx)) →→ PP((xx)))) is true iff, for some place a,is true iff, for some place a,
QQ((aa)) →→ PP((aa) is true.) is true.
QQ((aa)) →→ PP((aa)) is true iffis true iff
QQ((aa) is false) is false oror PP((aa) is true) is true
(conditional is only false in one row of table!)(conditional is only false in one row of table!)
So,So, the formula meansthe formula means “Some places are either“Some places are either
not free of charge or occupied”not free of charge or occupied”

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Module #1 - Logic
About the last of these
4.4. ∃∃xx ((QQ((xx)) →→ PP((xx)))) When confused by aWhen confused by a
conditional: re-write it usingconditional: re-write it using negationnegation andand
disjunctiondisjunction::
∃∃xx ((¬¬QQ((xx)) ∨∨ PP((xx)))) ⇔⇔ (p. 67)(p. 67)
∃∃xx ¬¬QQ((xx)) ∨∨ ∃∃x Px P((xx)) “Some places are“Some places are
not free of charge or some places arenot free of charge or some places are
occupied”occupied”

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Module #1 - Logic
• Combinations to remember:Combinations to remember:
1.1. ∃∃xx ((QQ((xx)) ∧∧ PP((xx))))
2.2. ∀∀xx ((QQ((xx)) →→PP((xx))))

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Module #1 - Logic
Natural language is ambiguous!
• ““Everybody likes somebody.”Everybody likes somebody.”
– For everybody, there is somebody they like,For everybody, there is somebody they like,
∀∀∀xx ∃∃yy LikesLikes((xx,,yy))
– or, there is somebody (a popular person) whomor, there is somebody (a popular person) whom
everyone likes?everyone likes?
∀∃∃yy ∀∀xx LikesLikes((xx,,yy))
• ““Somebody likes everybody.”Somebody likes everybody.”
– Same problem: Depends on context, emphasis.Same problem: Depends on context, emphasis.
[Probably more likely.]

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Module #1 - Logic
End of §1.3-1.4, Predicate Logic
• From these sections you should have learned:From these sections you should have learned:
– Predicate logic notation & conventionsPredicate logic notation & conventions
– Conversions: predicate logicConversions: predicate logic ↔↔ clear Englishclear English
– Meaning of quantifiers, equivalencesMeaning of quantifiers, equivalences
– Simple reasoning with quantifiersSimple reasoning with quantifiers
• Upcoming topics:Upcoming topics:
– Introduction to proof-writing.Introduction to proof-writing.
– Then: Set theory –Then: Set theory –
• a language for talking about collections of objects.a language for talking about collections of objects.

Nsu module 01-logic-final

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