lecture2

lecture2 - Discrete Computation Structures CSE 2353 Bhanu...

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Discrete Computation Structures CSE 2353 Bhanu Kapoor, PhD Department of Computer Science & Engineering Southern Methodist University, Dallas, TX bkapoor@smu.edu , 214-336-4973 September 01, 2009 Embrey 129, SMU, Dallas, Texas Lecture 02 1
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Lecture 02 Agenda ± Learn about sets ± Explore various operations on sets ± Venn diagrams epresentation of sets in computer memory ± Representation of sets in computer memory ± Learn about statements ± Logical connectives to combine statements ± Explore how to draw logical conclusions ± Quantifiers and predicates 2
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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 ² Examples: owels = {a e i o u} ± Vowels = {a,e,i,o,u} ± Primary colors = {red, blue, yellow} ± Membership examples ² “a belongs to the set of Vowels” is written as: a Vowels ² “j does not belong to the set of Vowels: j Vowels 3
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Sets ± Set-builder method ² 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 e property P the property P ± Example: ² 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} 4
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Sets ± Standard Symbols which denote sets of numbers ² N : The set of all natural numbers (i.e.,all positive integers) ² Z : The set of all integers ² Z* : The set of all nonzero integers ² E : The set of all even integers ² Q : The set of all rational numbers ² Q* : The set of all nonzero rational numbers ² Q + : The set of all positive rational numbers ² R : The set of all real numbers ² R* : The set of all nonzero real numbers ² R + : The set of all positive real numbers ² C : The set of all complex numbers ² C* : The set of all nonzero complex numbers 5
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Sets ± Subsets ² “X is a subset of Y” is written as X Y “ is not a subset of Y” is written as X Y ² X is not a subset of Y is written as X ² Example: ± X = {a,e,i,o,u}, Y = {a, i, u} and z = {b,c,d,f,g} ² Y X, since every element of Y is an element of X ² Y Z, since a Y, but a Z 6
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Sets ± Superset and Y are sets If X then “X is ² 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 ² X and Y are sets. X is a proper subset of Y if X and there exists at least one element in Y Y and there exists at least one element in Y that is not in X. This is written X Y. ² Example: ± X = {a,e,i,o,u}, Y = {a,e,i,o,u,y} ² X Y , since y Y, but y X 7
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Sets ± Set Equality ² X and Y are sets. They are said to be equal if every element of X is an element of Y and every element of Y is an element of X, i.e. X Y and Y X ² Examples: ± {1,2,3} = {2,3,1} = {red blue yellow} and Y = {c | c is a primary color} ± X = {red, blue, yellow} and Y = {c | c is a primary color} Therefore, X=Y ± Empty (Null) Set ² A Set is Empty (Null) if it contains no elements.
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This note was uploaded on 11/11/2009 for the course CSE 2353 taught by Professor Bhanukapoor during the Spring '09 term at SMU.

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lecture2 - Discrete Computation Structures CSE 2353 Bhanu...

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