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 < x < 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 (a[i] == b[j]) 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 Morgan’s 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. Russell’s 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.