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Discrete Mathematics
CSE 2353
Fall 2007
Margaret H. Dunham Margaret H. Dunham
Department of Computer Science and Engineering Department of Computer Science and Engineering
Southern Methodist University Southern Methodist University
- Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul,Some slides provided by Dr. Eric Gossett; Bethel University; St. Paul, Minnesota Minnesota
- Some slides are companion slides forSome slides are companion slides for Discrete Mathematical Structures:Discrete Mathematical Structures: Theory and Applications Theory and Applications by D.S. Malik and M.K. Senby D.S. Malik and M.K. Sen
2 Outline (^) Introduction (^) Sets Logic & Boolean Algebra Proof Techniques (^) Counting Principles (^) Combinatorics (^) Relations,Functions (^) Graphs/Trees Boolean Functions, Circuits Introduction Introduction
© Dr. Eric Gossett 4 The Stable Marriage Problem
In the future we will: (^) Prove that the assignment is stable (reading tonight). (^) Prove that the assignment is optimal for suitors. (^) Count the number of possible assignments. (^) Calculate the complexity of the algorithm.
The Problem A Solution: (^) The Deferred Acceptance Algorithm
© Discrete Mathematical Structures: Theory and Applications 5 Stable (^) Marriage partners should be assigned in such a manner that no one will be able to find someone (whom they prefer to their assigned mate) that is willing to elope with them.
7 (^) Introduction (^) Sets Logic & Boolean Algebra Proof Techniques (^) Counting Principles (^) Combinatorics (^) Relations,Functions (^) Graphs/Trees Boolean Functions, Circuits Outline Sets Sets
8
It is assumed that you have studied set theory before.
The remaining slides in this section are for your review. They will not all be covered in class.
If you need extra help in this area, a special help session will be scheduled.
© Discrete Mathematical Structures: Theory and Applications 10 Sets
Definition: Well-defined collection of distinct objects
Members or Elements: part of the collection
Roster Method: Description of a set by listing the
elements, enclosed with braces
Examples:
Vowels = {a,e,i,o,u}
Primary colors = {red, blue, yellow}
Membership examples
“a belongs to the set of Vowels” is written as:
a Vowels
“j does not belong to the set of Vowels:
j Vowels
© Discrete Mathematical Structures: Theory and Applications 11 Sets (^) Set-builder method (^) A = { x | x S, P(x) } or A = { x S | P(x) } (^) A is the set of all elements x of S, such that x satisfies the property P (^) Example: (^) If X = {2,4,6,8,10}, then in set-builder notation, X can be described as X = {n Z | n is even and 2 n 10}
© Discrete Mathematical Structures: Theory and Applications 13 Sets (^) Subsets (^) “X is a subset of Y” is written as X Y (^) “X is not a subset of Y” is written as X Y (^) Example:
X = {a,e,i,o,u}, Y = {a, i, u} and
Z= {b,c,d,f,g}
(^) Y X, since every element of Y is an element of X (^) Y Z, since a Y, but a Z
© Discrete Mathematical Structures: Theory and Applications 14 Sets (^) Superset (^) X and Y are sets. If X Y, then “X is contained in Y” or “Y contains X” or Y is a superset of X, written Y X Proper Subset (^) X and Y are sets. X is a proper subset of Y if X Y and there exists at least one element in Y that is not in X. This is written X Y. (^) Example: (^) X = {a,e,i,o,u}, Y = {a,e,i,o,u,y} (^) X Y , since y Y, but y X
© Discrete Mathematical Structures: Theory and Applications 16 Sets
Finite and Infinite Sets
X is a set. If there exists a nonnegative integer n such that X
has n elements, then X is called a finite set with n elements.
If a set is not finite, then it is an infinite set.
Examples:
Y = {1,2,3} is a finite set
P = {red, blue, yellow} is a finite set
E , the set of all even integers, is an infinite set
, the Empty Set, is a finite set with 0 elements
© Discrete Mathematical Structures: Theory and Applications 17 Sets
Cardinality of Sets
Let S be a finite set with n distinct elements, where
n ≥ 0. Then |S| = n , where the cardinality (number
of elements) of S is n
Example:
If P = {red, blue, yellow}, then |P| = 3
Singleton
A set with only one element is a singleton
Example:
H = { 4 }, |H| = 1, H is a singleton
© Discrete Mathematical Structures: Theory and Applications 19 Sets Venn Diagrams (^) Abstract visualization of a Universal set, U as a rectangle, with all subsets of U shown as circles. (^) Shaded portion represents the corresponding set (^) Example:
In Figure 1, Set X,
shaded, is a subset of
the Universal set, U
© Discrete Mathematical Structures: Theory and Applications 20 Set Operations and Venn Diagrams
Union of Sets
Example: If X = {1,2,3,4,5} and Y = {5,6,7,8,9}, then XUY = {1,2,3,4,5,6,7,8,9}
