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Unformatted text preview: CSE 1400 Applied Discrete Mathematics Sets Department of Computer Sciences College of Engineering Florida Tech Fall 2011 Set Basics 1 Common Sets 3 Operations On Sets 3 Precedence of Set Operations 4 Cartesian Products 4 Subset of a Set 5 Cardinality of a Set 6 Power Set of a Set 6 Binomial Coefficients 7 Counting Bit Strings 8 Boolean Laws 8 Partition of a Set 11 Venn and Euler Diagrams 12 Problems on Sets 15 Abstract Finite and countable sets are fundamental primitives of discrete math ematics. Operations can be defined on sets creating an “algebra.” Counting the number of elements in a set and counting subsets with a certain property are fundamental in computing probabilities and statistics. Partitioning a set describes equivalences among its elementss. Set Basics A set is an unordered collection of things . The things in a set A are said to be elements or members of A . The natural If a is an element in A write a ∈ A . Of course, it can occur that a particular element a is not a member of set A , in which case write a 6∈ A . cse 1400 applied discrete mathematics sets 2 number 7 is a member of the set O of octal numerals 7 ∈ O = { 0, 1, 2, 3, 4, 5, 6, 7 } The natural number 8 is not a member of the set O of octal numerals 8 6∈ O = { 0, 1, 2, 3, 4, 5, 6, 7 } A can be described by listing its members as a comma separated list enclosed in curly braces {} . No element is duplicated in a set: A thing in a set is listed once and only once. A can be also described For instance, B = { 0, 1 } , is the set of bits. by comprehension. For instance, B = { b : b is a bit } More generally, A can be comprehended by a description For instance, the set of even natural numbers is comprehended by the description 2 N = { a : a = 2 n ∧ n ∈ N } A = { a : p ( a ) is True } where p ( a ) is a proposition about variable a . Even more generally, A can be comprehended by a description For instance, the set of even natural numbers is comprehended by the description 2 N = { 2 a : a ∈ N } A = { f ( a ) : p ( a ) is True } where f is a function and p ( a ) is a proposition about the variable a . In computing practice, set comprehen sion requires the proposition p ( a ) to be computable, that is, there must be an algorithm that returns True when a ∈ A and False when a 6∈ A . When context demands it, the set of all possible things, called the universal set and denoted U , can be named. Sets can be For instance, the set of natural numbers is the universal set for many computing problems. Strings over an alphabet could be U in other applications. represented by diagrams, for instance, a single set X is drawn as a circle inside a rectangle. U X Two sets X and Y can be drawn in several relationships, called Euler diagrams. For instance, when no members of X are in Y the sets are disjoint ....
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 Fall '11
 Shoaff
 Math, Sets

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