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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 [email protected] , 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 Representation of sets in computer memory Learn about statements L i l i bi 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 Examples: Vowels = {a e i o u} 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 3
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Sets Set-builder method A { | S P( ) } A { S | P( ) } 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 { Z | i d 2 10} X = {n Z | n is even and 2 n 4
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Sets Standard Symbols which denote sets of numbers N : The set of all natural numbers (i.e.,all positive integers) Z Th t f ll i t Z : The set of all integers Z* : The set of all nonzero integers E : The set of all even integers Q : The set of all rational numbers Q* : The set of all nonzero rational numbers Q + : The set of all positive rational numbers R : The set of all real numbers R* : The set of all nonzero real numbers R + : The set of all positive real numbers C : The set of all complex numbers C* : The set of all nonzero complex numbers 5
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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 X is not a subset of Y is written as X 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 6
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Sets Superset X and Y are sets If X Y then “X is 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 7
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Sets Set Equality X and Y are sets. They are said to be equal if l t f X i l t f Y d 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 Examples: {1,2,3} = {2,3,1} X = {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 A Set is Empty (Null) if it contains no elements. The Empty Set is written as Th E t S t i b t f t The Empty Set is a subset of every set 8
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Sets Finite and Infinite Sets X i t If th i t ti 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.
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