I find the question, why are we here typically human'

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• I find the question, why are we here? typically
  human.
• Id suggest are we here? would bethe more
  logical choice.
• Leonard Nimoy
• Against logic there is no armor like ignorance.
• Laurence J. Peter
• Logic is like the sword those who live by it
• shall perish by it.
• Samuel Butler

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CSE 502NFundamentals of Computer Science

• Fall 2004
• Lecture 4
• Introduction to formal logic Boolean algebra
• (Rosen 1.1 1.4)

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Propositional Logic

• Proposition, p, is a declarative sentence that is
  either true or false, but not both
• Nairobi is the capital of Kenya (true
  proposition)
• 2 5 6 (false proposition)
• x 4 21 (not a proposition)
• Negation operator (Ø) produces a new proposition
  such that q Øp is the proposition it is not
  the case that p
• Note that p Ø(Øp)
• Formal definition of NOT operator or inverter
  logic gate

Truth Table
4
Connectives

• Conjuction (Ù) the proposition p Ù q (p and q)
  is true when both p and q are true and false
  otherwise
• Disjuction (Ú) the proposition p Ú q (p or q)
  is false when both p and q are false and true
  otherwise
• Exclusive-or (Å) the proposition p Å q (p xor
  q) is true when exactly one of p and q is true
  and false otherwise

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Implication

• Implication () the implication p q is the
  proposition that is false when p is true and q is
  false, and true otherwise
• An implication is a conditional statement
• p is the hypothesis (antecedent or premise)
• q is the conclusion (consequence)
• Equivalent statements if p, q, p only if q,
  q if p, q when p
• Examples
• If I am elected, then the economy will improve.
• I am elected only if the economy will improve.

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Converse, Contrapositive, Inverse

• Proposition q p is the converse of p q
• Proposition Øq Øp is the contrapositive of p
  q
• The contrapositive has the same truth value as
  the implication
• Proposition Øp Øq is the inverse of p q
• Converse and inverse are equivalent compound
  propositions

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Biconditional

• Biconditional () the biconditional p q is the
  proposition that is true when p and q have the
  same truth values
• Can also be thought of as Exclusive-NOR (XNOR),
  Ø(p Å q)
• Also equivalent to (p q) Ù (q p)
• Equivalent statements p if and only if q, p
  is necessary and sufficient for q, p if q
• The economy will improve if and only if I am
  elected.

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Precedence of Logical Operators

• Negation takes precedence over all other logical
  operators
• It is applied before all other operators
• Example Ø p Å q is the exclusive-or of Øp and q,
  i.e. (Øp) Å q
• Conjunction takes precedence over disjunction
• p Ù q Ú r means (p Ù q) Ú r
• Implication and biconditional operators have
  lower precedence than conjunction and disjunction
  operators

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Found in Translation

• You can drive your car on campus only if you are
  a graduate student or you are not a business
  student.
• p you can drive your car on campus
• q you are a graduate student
• r you are a business student
• p (q Ú Ør)

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Consistent Specifications

• Logic expressions are useful for translating
  natural language problem constraints into formal
  system specifications
• Example
• If specifications conflict, there is no way to
  develop a system that satisfies the constraints
• Propositional expressions are consistent if there
  is an assignment of truth values to the variables
  that makes all the expressions true
• Example Are the following propositions
  consistent?
• The checksum is correct and the timer has
  expired if only if the link has intermittent
  failures. p Ù q r
• If the checksum is correct, then the timer has
  expired and the link does not have intermittent
  failures or the timer has not expired and the
  link does have intermittent failures. p q Ù
  Ør Ú Øq Ù r

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Logical Equivalences

• Tautology a compound proposition that is always
  true, independent of the truth values of the
  constituent propositions
• Example p Ú Øp
• Contradiction a compound proposition that is
  always false, independent of the truth values of
  the constituent propositions
• Example p Ù Øp
• Contingency a compound proposition that is
  neither a tautology nor a contradiction
• Example p Å q
• Propositions p and q are logically equivalent, p
  º q, if p q is a tautology
• Example show that q Ù Ør Ú Øq Ù r º q Å r

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De Morgans Law

• Ø(p1 Ú p2 Ú Ú pn) º (Øp1 Ù Øp2 Ù Ù Øpn)
• Ø(p1 Ù p2 Ù Ù pn) º (Øp1 Ú Øp2 Ú Ú Øpn)
• Review other logical equivalences in Rosen 1.2
  (Tables 5, 6, 7)
• Examples
• p q º Øp Ú q
• Show that (p Ù q) (p Ú q) is a tautology