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2. Sets+Functions

# 2. Sets+Functions - MATH 224 Discrete Mathematics Basic...

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1 08/27/09 MATH 224 – Discrete Mathematics Basic Structures - Sets Sets provide a basis for many of the data structures used in computer science. As you already know sets are collections of any type of object without ordering and without duplicates. The original formalization of the set was developed by the mathematician Georg Cantor in the late 19 th century. Cantor is most famous for both his theory of sets and infinity. He developed the idea of a hierarchy of infinities with the smallest being countable infinity sets, e.g., the integers, followed what he called Aleph 1 ( א (1 the “smallest” uncountable set, e.g., the real numbers. His theory provides the basis for the concept that some problems are not solvable by computers.

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2 08/27/09 MATH 224 – Discrete Mathematics Set Examples Some examples of sets include: 1. The set of students in this class 2. The set N of natural number (all non-negative integers) {0, 1, 2, 3, …} 3. The set Z of all integers both positive and negative {…, -2, -1, 0, 1, 2, …} 4. The set Q of all rational numbers (numbers that can be expressed as p/q , where p and q are elements of Z 5. The set R of real numbers . 6. The set C of complex numbers. Which of these sets are countable and which are uncountable? Countable sets can also be finite. Can you name some other countable sets, both finite and infinite?
3 08/27/09 MATH 224 – Discrete Mathematics Basic Set Notation

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4 08/27/09 MATH 224 – Discrete Mathematics Sets and Venn Diagrams x ≥ 0 x Z Describe the sets represented by this Venn diagrams and there size. Solvable by N Solvable by computers Computational Problems Letters ASCII characters
5 08/27/09 MATH 224 – Discrete Mathematics Basic Set Notation 1. Vertical bars are used to indicate the size of a set or the number of elements in a set. So far example, the set of English lowercase letters L has size twenty six as indicated by |L| = 26 A set of this size can be put in one-to- one correspondence with the set {0, 1, 2, ..., 25}. 2. The power set of a set is indicated by P(X). This is the set of all subsets of X. So for example, if X = {a, b, c} then P(X) = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}} If a set has a finite number of elements, for example N, how many elements will its power set have?

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2. Sets+Functions - MATH 224 Discrete Mathematics Basic...

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