lecture 8 introduction to logic
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Lecture 8 Introduction to Logic. CSCI – 1900 Mathematics for Computer Science Spring 2014 Bill Pine. Lecture Introduction. Reading Kolman - Section 2.1 Logical Statements Logical Connectives / Compound Statements Negation Conjunction Disjunction Truth Tables Quantifiers - PowerPoint PPT PresentationTRANSCRIPT
Lecture 8Introduction to Logic
CSCI – 1900 Mathematics for Computer ScienceFall 2014Bill Pine
CSCI 1900 Lecture 8 - 2
Lecture Introduction
• Reading– Rosen - Section 1.1
• Logical Statements• Logical Connectives / Compound Statements
– Negation– Conjunction– Disjunction
• Truth Tables• Quantifiers
– Universal– Existential
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Statement of Proposition
• Proposition (statement of proposition) – a declarative sentence that is either true or false, but not both
• Examples:– Star Trek was a TV series: True– 2+3 = 5: True– Do you speak Klingon? This is a question, not a
statement
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Statement of Proposition (cont)
– x - 3 = 5: a declarative sentence, but not a statement since it is true or false depending on the value of x
– Take two aspirins: a command, not a statement– The current temperature on the surface of the
planet Venus is 800oF: a proposition of whose truth is unknown to us
– The sun will come out tomorrow: a proposition that is either true or false, but not both, although we will have to wait until tomorrow to determine the answer
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Notation
• x, y, z, … denote variables that can represent real numbers
• p, q, r,… denote propositional variables that can be replaced by statements– p: The sun is shining today– q: It is cold
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Negation (Not)
• If p is a statement, the negation of p is the statement not p
• Denoted ~p ( alternately: !p or p or p )• If p is True,
– ~p is False• If p is False,
– ~p is True• not is a unary operator for the collection of
statements and ~p is a statement if p is a statement
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Negation Truth Table
p ~ p
T F
F T
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Examples of Negation
• If p: 2+3 > 1 Then ~p: 2+3 < 1• If q: It is night Then
– ~q: It is not the case that it is night,• It is not night• It is day
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Conjunction
• If p and q are statements– The conjunction of p and q is the compound
statement “p and q”– Denoted p q
• p q is true only if both p and q are true• Example:
– p: ETSU parking permits are readily available– q: ETSU has plenty of parking – p q = ?
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Conjunction Truth Table
p q p q
T T T
T F F
F T F
F F F
English Conjunctives
• The following are common conjunctives in English:
And, Now, But, Still, So, Only, Therefore, Moreover, Besides, Consequently, Nevertheless, For, However, Hence, Both... And, Not only... but also, While, Then, So then
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Disjunction (Inclusive)
• If p and q are statements, – The (inclusive) disjunction of p and q is the compound
statement “p or q”– Denoted p q
• p q is true if either p is true or q is true or both are true
• Example:– p: I am a male q: I am under 90 years old– p q = ?– p: I am a male q: I am under 20 years old– p q = ?
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Disjunction (Inclusive) Truth Table
p q p q
T T T
T F T
F T T
F F F
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Exclusive Disjunction
• If p and q are statements– The exclusive disjunction true if either p is true or q
is true, but not both are true– Denoted p q
• Example:– p: It is daytime– q: It is night time– p q (in the exclusive sense) = ?
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Disjunction(Exclusive) Truth Table
p q p q
T T F
T F T
F T T
F F F
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Exclusive versus Inclusive
• Depending on the circumstances, some English disjunctions are inclusive and some of exclusive.
• Examples of Inclusive– “I have a dog” or “I have a cat”– “It is warm outside” or “It is raining”
• Examples of Exclusive– Today is either Tuesday or it is Thursday– The light is either on or off
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Compound Statements
• A compound statement is a statement made by joining simple propositions with logical connectors
• For n individual propositions, there are 2n possible combinations of truth values
• The truth table contains 2n rows identifying the truth values for the statement represented by the table
• Use parenthesis to denote order of precedence• has precedence over • Use parenthesis to ensure meaning is clear
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Truth Tables as Tools
• Compound statements can be easily and systematically investigated with truth tables
• Assign a portion of the compound statement to a column
• Final column represents the complete compound statement
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Compound Statement Example
p q p q ~p (p q) (~p)
T T T F T
T F F F F
F T F T T
F F F T T
(p q) (~p)
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Quantifiers
• Recall from Section 1.1, a set may be defined by its properties {x | P(x)}
• For a specific element e to be a member of the set, P(e) must evaluate to “true”
• P(x) is called a predicate or a propositional function
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Computer Science Functions
• if P(x) then execute certain steps
• while Q(x) do specified actions
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Universal Quantification
• Universal quantification of the predicate P(x) means “For all values of x, P(x) is true”
• Denoted x P(x)• The symbol is called the universal
quantifier• The order in which multiple universal
quantifications are applied does not matter (e.g., x y P(x,y) ≡ y x P(x,y) )
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Universal Examples:
• P(x): -(-x) = x– This predicate makes sense for all real numbers
x R– The universal quantification of P(x), x P(x),
is a true statement, because for all real numbers, -(-x) = x
• Q(x): x+1<4– x Q(x) is a false statement, because, for
example, Q(5) is not true
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Existential Quantification
• Existential quantification of a predicate P(x) is the statement:
“There exists a value of x for which P(x) is true.”
• Denoted x P(x)• Existential quantification may be applied to several
variables in a predicate• The order in which multiple existential
quantifications are considered does not matter
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Existential Examples:
• P(x): -(-x) = x– The existential quantification of P(x), x P(x), is a true
statement, because there is at least one real number where -(-x) = x
• Q(x): x+1<4– x Q(x) is a true statement, because, for example, Q(2)
is true• Nota Bene: is not the complement of
– An Example• N(x): x ≠ x• x N(x) is false and x N(x) is also false
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Applying Both and Quantifications
• Order of application matters• Example: Let A and B be n x n matrices• The statement A ( B | A + B = In )• Reads: for every A there is a B such that A + B = In• Prove by coming up for equations for bii and bij (ji)• Now reverse the order: B (A | A + B = In) • Reads: there exists a B, for all A, such that , A + B = In• The second proposition is F A L S E !
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Assigning Quantification - 1
• Let p: x R(x) [R(x): a person is good]– If p is true then x ~R(x) is false
• If every person is good, then there does not exist person who is not good
– If p is true then x R(x) is true• If every person is good, then there exists a person
who is good– If p is false then x ~R(x) is true
• If not every person is good, then there exists a person who is not good
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Assigning Quantification - 2
• Let p: x R(x) [R(x): a person is good]– If p is true then x ~R(x) is false
• If there exists a person who is good, then it is not true that all people are not good
– If p is false then x ~R(x) is true• If there does not exist a good person then every
person is not good
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Implications of the Previous Slide• Assume a statement is made that “for all x, P(x) is
true”– If we can find one case that is not true
• The statement is false– If there isn’t one case that is not true
• The statement is true• Example: positive integers, n,
n P(n) = n2 - n + 41 is a prime number – This is false because an integer resulting in a non-
prime value, i.e., n such that P(n) is false
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Key Concepts Summary
• Statement of Proposition• Logical Connectives / Compound Statements
– Negation– Conjunction– Disjunction
• Truth Tables• Quantifiers
– Universal– Existential