Introduction to Logic

1
Introduction to Logic
2
Learning Objectives

• Introduction to numbers
• Introduction to logic
• propositions
• propositional equivalences

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Introduction to Numbers

• Numbers N 0, 1, 2, Natural
  Numbers
• Z , -10, -9, -8, , 0, 1, 2,
  Integers
• Z 1, 2, 3, Positive Integers
• Q Rational Numbers
• R Real numbers
• Are there real irrational numbers?
• Notation a,b,c, n, k, m, i, j will usually
  denote integers (unless otherwise stated).

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Introduction to Numbers

• Binary operations - / (ab)
• The red operations are commutative ( a op b
  b op a ), the blue operations are not.
• The red operations are associative (a op b)
  op c a op (b op c) the blue are not.
• / is further distinct from the other
  operations a / b may not be an integer, all
  other operations when operated on integers yield
  an integer.
• Conclusion a / b / c is ambiguous. To resolve
  ambiguity we need parenthesis
• Either (a / b) / c or a / (b / c). For
  instance, if a 9, b c 3 then
• (a / b) / c 1 while a / (b / c) 9.

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Introduction to Numbers

• Math language n is an even integer n 2k
• n is an odd integer n 2k1 (or n ? 2k)
• c divides a a bc (c a)
• a is a multiple of c
• a is a square a n2
• The division algorithm For any given
  integer a and positive integer b
• a bq r where 0 ? r
• (q quotient, r remainder).
• (Note b a if and only if r 0).
• Examples a 37, b 8 37 84 5
  (q 4 and r 5)
• a -39, b 7 -39 7(-6) 3 (q
  -6 and r 3)

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Introduction to Numbers

• We expand our mathematical data base through
  Theorems.
• Examples
• Theorem 1.1 If n is odd then n2 is odd.
• Proof n is odd ? n 2k 1 n2 (2k1)2
  4k2 4k 1 2(2k2 2k) 1 2m 1.
  QED
• Theorem 1.2 The product of two consecutive
  integers is an even integer.
• Proof (in math language k(k1) 2m).
• Case 1 k is even, k 2j. k(k1) 2j(2j1)
  2(j(2j1)) 2m.
• Case 2 k is odd, k 2j1. k(k1)
  (2j1)(2j2) (2j1)2(j1) 2(j1)(2j1) 2m.
• QED

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Introduction to Numbers

• A positive integer p is prime if its only
  divisors are 1 and p. 2,3,5,7,11, are all
  prime numbers. If a is not prime it is called a
  composite number.
• There are infinitely many primes (Greeks more
  than 2200 years ago).
• Is everything known in Math? Anything left to be
  proved?
• Pairs of primes that differ by 2 such as (11,
  13), (17, 19) (29, 31) are called twin primes.
• Are three infinitely many twin primes?
• An open problem!!! Over 2000 years old!
• Maple introduction The functions isprime(n) and
  ithprime(n).

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Logic

• A proposition is a statement which is true or
  false but not both.
• Today is Saturday 2 25 11.
• Compound propositions
• Negation ?p (if p T then ?p F)
• Given two propositions, p, q. We can create new
  compound propositions.

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Logic

• Disjunction p ? q (p or q). This new
  proposition is False only when both p and q are
  False, for all other possibilities it is True.
• Conjunction p ? q (p and q). This new
  proposition is True only when both p and q are
  True, for all other possibilities it is False.
• Implication p ? q This new proposition is
  False only when p is True and q is False,
  otherwise it is True. p is called the hypothesis
  (antecedent, premise), and q the conclusion (or
  consequence).
• XOR (exclusive or) p ? q This new
  proposition is True only when either p or q are
  True, but not both. It is False otherwise.

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Truth Tables
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Truth Tables
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Logic

• Example
• Five friends have access to a chat room. Is it
  possible to determine who is chatting if the
  following information is known? Either Kevin or
  Heather or both, are chatting. Either Randy or
  Vijay, but not both are chatting. If Abby is
  chatting, so is Randy. Vijay and Kevin are
  either both chatting or neither are chatting. If
  Heather is chatting, then so are Abby and Kevin.

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Logic

• Ans. Create five propositions A Abbey is
  chatting.
• H Heather is chatting
• R Randy is chatting
• V Vijay is chatting
• K Kevin is chatting

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Logic

• Either Kevin or Heather or both, are chatting
  (1) H ? K is True.
• Either Randy or Vijay, but not both are
  chatting (2) (R ? V) is True.
• If Abby is chatting, so is Randy (3) A ? R
  is True
• Vijay and Kevin are either both chatting or
  neither are chatting
• (4) ( K ? V) ? (?K ? ?V) is True
• If Heather is chatting, then so are Abby and
  Kevin. (5) H ? A ? K is True.

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Logic

• Assume H True.
• From (5) A K True.
• From (3) R True.
• From (2) V False but then (4) is False!!!
  Contradiction.
• So H False.
• From (1) K True.
• From (4) V True.
• From (2) R False.
• From (3) A False.
• Answer YES!!! Vijay and Kevin are chatting!!!

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Logic

• Q ? P is the CONVERSE of P ? Q
• ? Q ? ? P is the CONTRAPOSITIVE of P ? Q
• _______________
• Example
• Find the converse and contrapositive of the
  following
• statement
• R Raining tomorrow is a sufficient condition
  for my not
• going to town.
• Step 1 Assign propositional variables to
  component
• propositions
• P It will rain tomorrow
• Q I will not go to town

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Logic

• Step 2 Symbolize the assertion
• R P ? Q
• Step 3 Symbolize the converse
• Q ? P
• Step 4 Convert the symbols back into words
• If I dont go to town then it will rain
  tomorrow
• or
• Raining tomorrow is a necessary condition for my
• not going to town.
• or
• My not going to town is a sufficient condition
  for it
• raining tomorrow.

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Logic

• Biconditional
• if and only if, iff
• Symbol ?
• _________________
• Example P - I am going to town, Q - It is
  going to rain
• P ? Q I am going to town if and only if it is
  going to
• rain.
• Truth Table
• Note Both P and Q must have the same truth
  value.
• P Q P ? Q
• 0 0 1
• 0 1 0
• 1 0 0
• 1 1 1

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Propositional Equivalences

• A tautology is a proposition which is always true
  .Classic Example P ? ? P
• ___________________
• A contradiction is a proposition which is always
  false .Classic Example P ? ? P
• ___________________
• A contingency is a proposition which neither a
  tautology
• nor a contradiction.Example (P ? Q) ? ? R
• ____________________

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Propositional Equivalences

• Two propositions P and Q are logically equivalent
  if P ? Q is a tautology.
• We write P?Q
• ____________________
• Example show that (P ? Q) ? (Q ? P) and (P ? Q)
  are logically equivalent (P ? Q) ? (Q ? P) ?(P ?
  Q)

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Propositional Equivalences
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Propositional Equivalence

• To show that two propositions are NOT logically
  equivalent, it is enough to show a counterexample
  where both propositions do not have the same
  truth value.

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Propositional Equivalence

• Show that (p ? q) ? r and p ? (q ? r ) are NOT
  logically equivalent.
• Note it is enough to show that for one
  assignment of truth values to p, q and r the two
  proposition will have distinct truth values.
• Let p F, q T and r F.
• Then (p ? q) ? r (F ? T) ? F T ? F
  F
• p ? (q ? r) F ? (T ? F) F ? F T