H-SC MATH 262 - Lecture 24 - Basic Definitions of Set Theory

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Basic Definitions of Set TheoryThe Universal SetSet OperationsSets and Boolean OperatorsSlide 5Set DifferencesSlide 7SubsetsSlide 9Sets Defined by a PredicateSlide 11Intersection and UnionComplements and DifferencesSlide 14Basic Definitions of Set TheoryLecture 24Section 5.1Fri, Mar 2, 2007The Universal SetWhenever we use sets, there must be a universal set U which contains all elements under consideration.Typical examples are U = R and U = N.Without a universal set, taking complements of set is problematic.Set OperationsLet A and B be set.Define the intersection of A and B to beA  B = {x  U | x  A and x  B}.Define the union of A and B to beA  B = {x  U | x  A or x  B}.Define the complement of A to beAc = {x  U | x  A}.Sets and Boolean OperatorsA set may be represented as a sequence of true and false values.Let the universal set be U = {a1, a2, a3, …}.Then the set A = {a1, a3, …} could be represented as {T, F, T, …} or as {1, 0, 1, …}.Sets and Boolean OperatorsWhat boolean operations correspond to the set operations of union, intersection, and complementation?Set DifferencesDefine the difference A minus B to beA – B = {x  U | x  A and x  B}.Define the symmetric difference of A and B to beA  B = (A – B)  (B – A).Sets and Boolean OperatorsWhat boolean operations correspond to the set operations of difference and symmetric difference?SubsetsA is a subset of B, written A  B, if x  A, x  B.A equals B, written A = B, if x  A, x  B and x  B, x  A.A is a proper subset of B, written A  B, if x  A, x  B and x  B, x  A.Sets and Boolean OperatorsIs there a boolean operator that corresponds to the subset relation?That is, an operation * on boolean variables such that A*B is true if and only if A  B?Sets Defined by a PredicateLet P(x) be a predicate.Define a set A = {x  U | P(x)}.For any x  U,If P(x) is true, then x  A.If P(x) is false, then x  A.A is the truth set of P(x).Sets Defined by a PredicateTwo special cases.What predicate defines the universal set?What predicate defines the empty set?Intersection and UnionLet P(x) and Q(x) be predicates and defineA = {x  U | P(x)}.B = {x  U | Q(x)}.Then the intersection of A and B isA  B = {x  U | P(x)  Q(x)}.The union of A and B isA  B = {x  U | P(x)  Q(x)}.Complements and DifferencesThe complement of A isAc = {x  U | P(x)}.The difference A minus B isA – B = {x  U | P(x)  Q(x)}.The symmetric difference of A and B isA  B = {x  U | P(x)  Q(x)}.SubsetsA is a subset of B if x  U, P(x)  Q(x), orx  A, Q(x).A equals B if x  U, P(x)  Q(x), orx  A, Q(x) and x  B, P(x).A is a proper subset of B if x  A, Q(x) and x  B,


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