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Unformatted text preview: [Lecture 06][Set Theory] Discrete Structures 1 [Lecture 06][Set Theory] TDS1191 Discrete Structures Lecture 06 Set Theory Multimedia University Trimester 2, Session 2011/2012 [Lecture 06][Set Theory] Discrete Structures 2 [Introduction] Set, Elements Equal, Cardinality Subset, Power Set [Set Operations] Union, Intersection, Difference, Complement Set Identities [Lecture 06][Set Theory] Discrete Structures 3 Basic Notations/Properties of Sets A set is simply an unordered collection of objects/elements/members. Sets are usually denoted as follows S = {a, b, c} T = {x  P(x)} where P is a predicate x S (x is an element of S) is the proposition that object x is an element or a member of the set S. Examples: 3 N if N = {0, 1, 2, 3, 4, } a T if T = {x  x is a letter of the English alphabets} x S : (x S) x is not in S Exercise: What is the truth value of each of the following propositions? i.{4} {x  x is an odd number} T/F ii.e {x  x is a vowel} T/F iii.cat {x  x is an animal} T/F iv.Peter {x  x is a good boy} T/F An example of a predicate: P(x) : x is a course offered by FIT Exercise: What is the truth value of each of the following propositions? i.{4} {x  x is an odd number} T/ F ii.e {x  x is a vowel} T /F iii.cat {x  x is an animal} T /F iv.Peter {x  x is a good boy} T/F [Lecture 06][Set Theory] Discrete Structures 4 Basic Notations/Properties of Sets Common Sets to Remember N = {0, 1, 2, 3, 4, 5, } Set of natural numbers Z = {,  2,  1, 0, 1, 2, } Set of integers Z + = {1, 2, 3, } Set of positive integers Q = { p / q  p Z, q Z, q 0} Set of rational numbers R = {All real numbers} Set of real numbers Examples of real numbers:12.66547, 100000000.02, 244.0, Another name for rational number is fraction 2 Another name for natural number is counting number [Lecture 06][Set Theory] Discrete Structures 5 Basic Notations/Properties of Sets II S = T if and only if ( 2200 x : x S x T) Exercise: Given M 1 = { x  x is an integer where x >0 and x <5 } M 2 = { x  x is a positive integer whose square is >0 and <25} Decide whether M 1 = M 2. Hence, two sets are equal iff they have the same members. It does not matter how the set is defined or denoted! Set equality This is read as: For all x , x is an element of set S if and only if x is an element of set T. [Lecture 06][Set Theory] Discrete Structures 6 Basic Notations/Properties of Sets III Exercise: {cat,rabbit,parrot} = ____ {a,b} = ___ {{1,2,3},{4,5}} = ____ { x  x is even and x < 11} = ___ Can you think of some examples of infinite sets?...
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 Spring '11
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