Logic

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Section 1.1

• Logic

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Proposition

• Statement that is either true or false
• cant be both
• in English, must contain a form of to be
• Examples
• Cate Sheller is President of the United States
• CS1 is a prerequisite for this class
• I am breathing

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Many statements are not propositions ...

• Give me liberty or give me death
• ax2 bx c 0
• See Spot run
• Who am I and why am I here?

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Representing propositions

• Can use letter to represent proposition think of
  letter as logical variable
• Typically use p to represent first proposition, q
  for second, r for third, etc.
• Truth value of a proposition is T (true) or
  F(false)

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Negation

• Logical opposite of a proposition
• If p is a proposition, not p is its negation
• Not p is usually denoted
• ?p

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Truth table

• Graphical display of relationships between truth
  values of propositions
• Shows all possible values of propositions, or
  combinations of propositions

p ?p T F F T
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Logical Operators

• Negation is an example of a logical operation
  the negation operator is unary, meaning it
  operates on one logical variable (like unary
  arithmetic negation)
• Connectives are operators that operate on two (or
  more) propositions

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Conjunction

• Conjunction of 2 propositions is true if and only
  if both propositions are true
• Denoted with the symbol ?
• If p and q are propositions, p ? q means p AND q
• Remember - ? looks like A for And

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Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
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Truth table for p ? q
p q p ? q T T T T F F F T F F F
F
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Disjunction

• Disjunction of two propositions is false only if
  both propositions are false
• Denoted with this symbol ?
• If p and q are propositions, p ? q means p OR q
• Mnemonic ? looks like OAR in the water (sort of)

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Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
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Truth table for disjunction
p q p ? q T T T T F T F T T F F
F
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Inclusive vs. exclusive OR

• Disjunction means or in the inclusive sense
  includes the possibility that both propositions
  are true, and can be true at the same time
• For example, you may take this class if you have
  taken Calculus I or you have the instructors
  permission - in other words, you can take it if
  you have either, or both

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Exclusive OR

• The exclusive or of two propositions is true when
  exactly one of the propositions is true, false
  otherwise
• Exclusive or is denoted with this symbol ?
• For p and q, p ? q means p XOR q
• Mnemonic ? looks like sideways X inside an O

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English examples

• I am either in class or in my office
• The meal comes with soup or salad
• You can have your cake or you can eat it

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Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
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Truth table for ?
p q p ? q T T F T F T F T T F F
F
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Implication

• The implication of two propositions depends on
  the ordering of the propositions
• The first proposition is calls the premise (or
  hypothesis or antecedent) and the second is the
  conclusion (or consequence)
• An implication is false when the premise is true
  but the conclusion is false, and true in all
  other cases

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Implication

• Implication is denoted with the symbol ?
• For p and q, p ? q can be read as
• if p then q
• p implies q
• q if p
• p only if q
• q whenever p
• q is necessary for p
• p is sufficient for q
• if p, q

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Implication

• Note that a false premise always leads to a true
  implication, regardless of the truth value of the
  conclusion
• Implication does not necessarily mean a cause and
  effect relationship between the premise and the
  conclusion

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Implications in English

• If Cate lives in Iowa, then Discrete Math is a
  3-credit class
• Since p (I live in Iowa) and q (this is a
  3-credit class) are both true, p ? q is true even
  though p and q are unrelated statements

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Implications in English

• If the sky is brown, then 225
• Since p (sky is brown) and q (225) are both
  false, the implication p ? q is true
• Remember, you can conclude anything from a false
  premise

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If/then vs. implication

• In programming, the if/then logic structure is
  not the same as implication, though the two are
  related
• In a program, if the premise (if expression) is
  true, the statements following the premise will
  executed, otherwise not
• There is no conclusion, so its not an
  implication

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Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
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Truth table for ?
p q p ? q T T T T F F F T T F F
T
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Converse contrapositive

• For the implication p ? q, the converse is q ? p
• For the implication p ? q, the contrapositive is
  ?q ? ?p

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Biconditional

• A biconditional is a proposition that is true
  when p and q have the same truth values (both
  true or both false)
• For p and q, the biconditional is denoted as p ?
  q, which can be read as
• p if and only if q
• p is necessary and sufficient for q
• if p then q, and conversely

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Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of p ? q
p ? r r ? q ?p ? r ?p ? ?r ?(p ? r)
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Truth table for ?
p q p ? q T T T T F F F T F F F
T
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Compound propositions

• Can build compound propositions by combining
  simple propositions using negation and
  connectives
• Use parentheses to specify order or operations
• Negation takes precedence over connectives

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Examples
Let p 2 2 4 q It is raining r I
am in class now What is the value of (p ? q)
?? ( p ? r) (r ? q) ? (?p ? r) ?(p ? ?r ) ?
?(p ? r)
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Logic Bit Operations

• A bit string is a sequence of 1s and 0s - the
  number of bits in the string is the length of the
  string
• Bit operations correspond to logical operations
  with 1 representing T and 0 representing F

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Bit operation examples
Let s1 10011100 s2 11000110 s1 OR s2
11011110 s1 AND s2 10000100 s1 XOR s2
01011010
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Section 1.1

• Logic
• - ends -