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Discrete Mathematics Lecture 5 Alexander Bukharovich New York University

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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

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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

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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

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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

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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++; }

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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

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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

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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?

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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

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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

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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

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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

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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

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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?

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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.

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