Discrete Mathematics Lecture 5

1
Discrete MathematicsLecture 5
Alexander Bukharovich New York University
2
Basics of Set Theory

• Set and element are undefined notions in the set
  theory and are taken for granted
• Set notation 1, 2, 3, 1, 2, 3, 1, 2,
  3, 1, 2, 3, , ?, x ? R -3 lt x lt 6
• Set A is called a subset of set B, written as A ?
  B, when ?x, x ? A ? x ? B
• A is a proper subset of B, when A is a subset of
  B and ?x ? B and x ? A
• Visual representation of the sets
• Distinction between ? and ?

3
Set Operations

• Set a equals set B, iff every element of set A is
  in set B and vice versa
• Proof technique for showing sets equality
• Union of two sets is a set of all elements that
  belong to at least one of the sets
• Intersection of two sets is a set of all elements
  that belong to both sets
• Difference of two sets is a set of elements in
  one set, but not the other
• Complement of a set is a difference between
  universal set and a given set

4
Cartesian Products

• Ordered n-tuple is a set of ordered n elements.
  Equality of n-tuples
• Cartesian product of n sets is a set of n-tuples,
  where each element in the n-tuple belongs to the
  respective set participating in the product

5
Formal Languages

• Alphabet ? set of characters used to construct
  strings
• String over alphabet ? either empty string of
  n-tuple of elements from ?, for any n
• Length of a string is value n
• Language is a subset of all strings over ?
• ?n is a set of strings with length n over ?
• ? is a set of all strings of finite length over
  ?
• Notation for arithmetic expressions prefix,
  infix, postfix

6
Subset Check Algorithm

• Let two sets be represented as arrays A and B
• m size of A, n size of B
• i 1, answer yes
• while (i ? m answer yes)
• j 1, found no
• while (j ? n found no)
• if (ai bj) found yes
• j
• if (found no) answer no
• i

7
Set Properties

• Inclusion of Intersection
• A ? B ? A and A ? B ? B
• Inclusion in Union
• A ? A ? B and B ? A ? B
• Transitivity of Inclusion
• (A ? B ? B ? C) ? A ? C
• Set Definitions
• x ? X ? Y ? x ? X ? y ? Y
• x ? X ? Y ? x ? X ? y ? Y
• x ? X Y ? x ? X ? y ? Y
• x ? Xc ? x ? X
• (x, y) ? X ? Y ? x ? X ? y ? Y

8
Set Identities

• Commutative Laws A ? B A ? B and A ? B B ? A
• Associative Laws (A ? B) ? C A ? (B ? C) and
  (A ? B) ? C A ? (B ? C)
• Distributive Laws
• A ? (B ? C) (A ? B) ? (A ? C) and A ? (B ? C)
  (A ? B) ? (A ? C)
• Intersection and Union with universal set A ? U
  A and A ? U U
• Double Complement Law (Ac)c A
• Idempotent Laws A ? A A and A ? A A
• De Morgans Laws (A ? B)c Ac ? Bc and (A ? B)c
  Ac ? Bc
• Absorption Laws A ? (A ? B) A and A ? (A ? B)
  A
• Alternate Representation for Difference A B
  A ? Bc
• Intersection and Union with a subset if A ? B,
  then A ? B A and A ? B B

9
Exercises

• Is is true that (A B) ? (B C) A C?
• Show that (A ? B) C (A C) ? (B C)
• Is it true that A (B C) (A B) C?
• Is it true that (A B) ? (A ? B) A?

10
Empty Set

• S x ? R, x2 -1
• X 1, 3, Y 2, 4, C X ? Y
• Empty set has no elements ?
• Empty set is a subset of any set
• There is exactly one empty set
• Properties of empty set
• A ? ? A, A ? ? ?
• A ? Ac ?, A ? Ac U
• Uc ?, ?c U

11
Set Partitioning

• Two sets are called disjoint if they have no
  elements in common
• Theorem A B and B are disjoint
• A collection of sets A1, A2, , An is called
  mutually disjoint when any pair of sets from this
  collection is disjoint
• A collection of non-empty sets A1, A2, , An is
  called a partition of a set A when the union of
  these sets is A and this collection consists of
  mutually disjoint sets

12
Power Set

• Power set of A is the set of all subsets of A
• Theorem if A ? B, then P(A) ? P(B)
• Theorem If set X has n elements, then P(X) has
  2n elements

13
Boolean Algebra

• Boolean Algebra is a set of elements together
  with two operations denoted as and and
  satisfying the following properties
• a b b a, a b b a
• (a b) c a (b c), (a b) c a (b
  c)
• a (b c) (a b) (a c), a (b c) (a
  b) (a c)
• a 0 a, a 1 a for some distinct unique 0
  and 1
• a ã 1, a ã 0

14
Exercises

• Simplify A ? ((B ? Ac) ? Bc)
• Symmetric Difference A ? B (A B) ? (B A)
• Show that symmetric difference is associative
• Are A B and B C necessarily disjoint?
• Are A B and C B necessarily disjoint?
• Let S 2, 3, , n. For each Si ? S, let Pi be
  the product of elements in Si. Show that
• ?Pi (n 1)! / 2 1

15
Russells Paradox

• Set of all integers, set of all abstract ideas
• Consider S A, A is a set and A ? A
• Is S an element of S?
• Barber puzzle a male barber shaves all those men
  who do not shave themselves. Does the barber
  shave himself?
• Consider S A ? U, A ? A. Is S ? S?

16
Halting Problem

• There is no computer algorithm that will accept
  any algorithm X and data set D as input and then
  will output halts or loops forever to
  indicate whether X terminates in a finite number
  of steps when X is run with data set D.