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# Lecture 8 Introduction to Logic - PowerPoint PPT Presentation

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

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### Lecture 8Introduction to Logic

CSCI – 1900 Mathematics for Computer Science

Spring 2014

Bill Pine

• Kolman - Section 2.1

• Logical Statements

• Logical Connectives / Compound Statements

• Negation

• Conjunction

• Disjunction

• Truth Tables

• Quantifiers

• Universal

• Existential

CSCI 1900

• 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

CSCI 1900

• 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

CSCI 1900

• 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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• 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

CSCI 1900

CSCI 1900

• 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

CSCI 1900

• 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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• 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

CSCI 1900

• 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 = ?

CSCI 1900

CSCI 1900

• 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) = ?

CSCI 1900

CSCI 1900

• 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

CSCI 1900

• 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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• 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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(pq) (~p)

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• 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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• if P(x) then

execute certain steps

• while Q(x)

do specified actions

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• 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) )

CSCI 1900

• 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

CSCI 1900

• 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

CSCI 1900

• 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 (ji)

• 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 !

CSCI 1900

• 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

CSCI 1900

• 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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• 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

CSCI 1900

• Statement of Proposition

• Logical Connectives / Compound Statements

• Negation

• Conjunction

• Disjunction

• Truth Tables

• Quantifiers

• Universal

• Existential

CSCI 1900