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

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### Lecture 8Introduction to Logic

CSCI – 1900 Mathematics for Computer Science

Spring 2014

Bill Pine

Lecture Introduction
• Kolman - Section 2.1
• Logical Statements
• Logical Connectives / Compound Statements
• Negation
• Conjunction
• Disjunction
• Truth Tables
• Quantifiers
• Universal
• Existential

CSCI 1900

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)
• 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
• 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)
• 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
• 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
• 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
• 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)
• 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
• 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
• 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
• 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
• 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

(pq) (~p)

CSCI 1900

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
• if P(x) then

execute certain steps

• while Q(x)

do specified actions

CSCI 1900

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:
• 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 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:
• 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
• 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
• 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
• 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
• 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
• Statement of Proposition
• Logical Connectives / Compound Statements
• Negation
• Conjunction
• Disjunction
• Truth Tables
• Quantifiers
• Universal
• Existential

CSCI 1900