Sets - CSE 1400 Applied Discrete Mathematics Sets...

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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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Sets - CSE 1400 Applied Discrete Mathematics Sets...

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