Discrete Mathematics Lecture 5
Discrete Mathematics Lecture 5. Alexander Bukharovich New York University. 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}
Discrete Mathematics Lecture 5
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Discrete MathematicsLecture 5 Alexander Bukharovich New York University
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
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
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
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
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++; }
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
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
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?
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
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
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
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
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
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?
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.
