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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

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Lecture 8 introduction to logic

Lecture 8Introduction to Logic

CSCI – 1900 Mathematics for Computer Science

Spring 2014

Bill Pine


Lecture introduction
Lecture Introduction

  • Reading

    • Kolman - Section 2.1

  • Logical Statements

  • Logical Connectives / Compound Statements

    • Negation

    • Conjunction

    • Disjunction

  • Truth Tables

  • Quantifiers

    • Universal

    • Existential

CSCI 1900


Statement of proposition
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

CSCI 1900


Statement of proposition cont
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

CSCI 1900


Notation
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

CSCI 1900


Negation not
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

CSCI 1900



Examples of negation
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

CSCI 1900


Conjunction
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 = ?

CSCI 1900



English conjunctives
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

CSCI 1900


Disjunction inclusive
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 = ?

CSCI 1900



Exclusive disjunction
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) = ?

CSCI 1900



Exclusive versus inclusive
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

CSCI 1900


Compound statements
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

CSCI 1900


Truth tables as tools
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

CSCI 1900


Compound statement example
Compound Statement Example

(pq) (~p)

CSCI 1900


Quantifiers
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

CSCI 1900


Computer science functions
Computer Science Functions

  • if P(x) then

    execute certain steps

  • while Q(x)

    do specified actions

CSCI 1900


Universal quantification
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) )

CSCI 1900


Universal examples
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

CSCI 1900


Existential quantification
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

CSCI 1900


Existential examples
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

CSCI 1900


Applying both and quantifications
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


Assigning quantification 1
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

CSCI 1900


Assigning quantification 2
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

CSCI 1900


Implications of the previous slide
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

CSCI 1900


Key concepts summary
Key Concepts Summary

  • Statement of Proposition

  • Logical Connectives / Compound Statements

    • Negation

    • Conjunction

    • Disjunction

  • Truth Tables

  • Quantifiers

    • Universal

    • Existential

CSCI 1900


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