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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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