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Lecture Introduction

- Reading
- Kolman - Section 2.1
- 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

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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

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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

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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

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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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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., xy P(x,y) ≡ yx 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

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