ISA 780 Part 2

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ISA 780 Part 2

• Introductory Logic

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Instructor information

• Name Duminda Wijesekera
• Office hours 3.00-4.00 pm, Monday and Tuesdays
• Office ST II room 351.
• Mail box location ST II room 330.
• Email dwijesek_at_gmu.edu
• Telephone [phone]
• Teaching assistant None!

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Course Content

• A mathematical introduction to logic
• Objective Learn first order logic
• Expectation Learn proofs!
• Basic material
• Propositional logic
• Predicate logic
• Compactness theorem
• Consistency and completeness theorem
• Incompleteness theorem of G?del

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Course administration

• Textbook
• A mathematical introduction to logic
• Herbert Enderton
• Second edition ISBN0-12-238452-0
• The author has a web page at http//www.math.ucla.
  edu/hbe/amil/index.html
• What do we cover?
• Start with chapter 0 and cover as much as
  possible
• Main emphasis is Chapter 2. (all of it in detail!)

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Course evaluation

• Will assign homework daily
• All homework due in two weeks, in class, in my
  mailbox or under my door.
• Exam on the last day of class
• Exam counts for 20 of the total grade in ISA 780
• Homework count for another 30, making this part
  worth 50 of your ISA 780 grade
• A proof is given credit only if it is correct.
• Partial credit is given only if the partial proof
  can be extended to a complete proof of the claim.

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Other comments

• May not always be able to have power-point
  transparencies.
• Microsoft may not provide all the symbols!
• The students are expected to READ the appropriate
  section of the chapter in its entirety.
• University, ITE and departmental standards of
  ethical behavior applies.
• Not a standard ISA course. No Alice and Bob
  stories!
• That is, no security is discussed here.
• Similar courses are taught in computer science
  and Mathematics departments every where. See
  Endertons course in UCLA, introduction to
  mathematical logic at Cornells Math Department
  etc.

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Chapter 0

• Useful facts about sets and counting

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Review of set notation -1

• x? y means x is a member of the set y
• x? y means x is not a member of the set y
• The denial of the former
• Similar notation for equality with and ?
• Sets are extensional. That is two set A and B are
  said to be equal (written AB) iff
• for every x, x?A iff x?B

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Review of set notation -2

• Subset A ?B iff for every x, x?A implies x?B
• Power set P(A) B B ? A
• Known set operations on sets U, n, \
• Special set Ø the empty set
• Means x?Ø is false for any x.
• For collections of sets A (that is, their members
  are sets) UA, nA, are the union and intersection
  of all their members
• Example Let A0,1,5,1,6,1,5
• UA 0,1,5,6
• nA1

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Review of set notation-3

• Ordered pair ltx,ygtx,x,y
• First exercise Show that ltx,ygtlta,bgt iff xa and
  yb
• Extending notation ltx,y,zgtltltx,ygt,zgt
• General definition ltx1,x2,..xn,xn1gtltltx1,..xngt,x
  n1gt
• Lemma 0A(page 4)

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Lemma 0A (page 04)

• If ltx1,x2,..xmgtlty1,..ym,ymkgt, then
  x1lty1,..,yk1gt.
• Proof Follow the inductive argument in the
  textbook

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Functions and relations

• Cartesian product
• AxBlta,bgta?A, b?B
• Relation R from A to B is a subset of AxB.
• Say R(a,b) iff (a,b) ?R
• Domain(R) a ?b ?B (a,b) ?R
• Range(R) b ?a ?A (a,b) ?R
• A function is a single valued relation.
• That is, if R(x,y) and R(x,z) then yz.
• Notation R(x) is used for y.

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Equivalence relations

• A binary relation R is said to be
• Reflexive iff R(x,x) for all x.
• Symmetric iff R(x,y) implies R(y,x) for all x,y
• Transitive If R(x,y) and R(y,z) then R(x,z) for
  all x,y and z.
• An equivalence relation a reflexive, symmetric
  and transitive relation

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Equivalence classes

• Suppose R is an equivalence relation. Then for
  every x?Domain(R) define,
• xy?Domain(R) R(x,y), said to be the
  equivalence class of x under R.
• Lemma
• xy iff R(x,y)
• xy or xnyØ
• If x?Domain(R), then x?y for some y
• Proof

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Counting

• Notation N is the set of natural numbers.
• A set A is said to be countable if there is a
  one-to-one function (say) f from A to N.
• That is, f A ? N

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Counting - 2

• Can make the above function (say the counting
  function) ONTO.
• How?
• Let a0 be chosen so that f(a0)min Range(F)
• Let f(a0)0
• Suppose f(a0), ..f(an) has been defined to
  satisfy f(ai)I for all iltn. Then choose an1 to
  satisfy f(x) min Range(f) \ f(ai) iltn.
• Let f(an1)n1.

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Counting 3

• Lemma 0B If A is countable, then the set of all
  finite sequences of A are countable.
• Proof

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Zorns Lemma

• A chain of sets A collection of sets C is said
  to be a chain iff for any two sets c,c? C
  satisfy c?c or c?c.
• Zorns Lemma If a collection A of sets satisfy
  the condition that for any chain C ? A, then U C
  ? A, then A has a maximal element.
• That is there is a set M?A satisfying the
  condition that M?M ?A satisfy MM

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Cardinal Numbers

• Tow sets A and B are equinumerous iff there is a
  bijection f A?B.
• Equinumerousness is an equivalence relation.
• For finite sets, we use natural numbers to count
  the number of elements in them. Cardinality
  extends this to infinite sets.

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Dominating sets

• We say a set A is dominated by a set B (written A
  B) iff A is equinumerous to a subset of B.
• Schr?der-Bernstein theorem If A B and B A,
  then there is a bijection f A?B.
• Will return to defining cardinals if they are
  needed later on.