1. Discrete Mathematics Lecture 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  X c  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: (A c ) c = A Idempotent Laws: A  A = A and A  A = A De Morgan’s Laws: (A  B) c = A c  B c and (A  B) c = A c  B c Absorption Laws: A  (A  B) = A and A  (A  B) = A Alternate Representation for Difference: A – B = A  B c 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, x 2 = -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  A c = , A  A c = U –U c = ,  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 A 1, A 2, …, A n is called mutually disjoint when any pair of sets from this collection is disjoint A collection of non-empty sets {A 1, A 2, …, A n } 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 2 n 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  A c )  B c ) 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 S i  S, let P i be the product of elements in S i. Show that:  P i = (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.