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Sets

Sets. Model Notation Sub-Sets Operations (abstract) Operations (ADT) Specialized ADTs based on Set. Sets – the Model. Set: a collection of elements (members) without repetition {a, b, c, ... } Usually, there is a linear order specified on the members specified somewhere else!

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Sets

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  1. Sets Model Notation Sub-Sets Operations (abstract) Operations (ADT) Specialized ADTs based on Set CS 303 – Sets Lecture 8

  2. Sets – the Model • Set: a collection of elements (members) without repetition • {a, b, c, ... } • Usually, there is a linear order specified on the members • specified somewhere else! • not based on “position” • Well Defined: if a,b S, then either a < b, a = b, or b < a • Transitive: if a,b,c S, then (a<b) and (b<c) implies (a<c) CS 303 – Sets Lecture 8

  3. Set Notation • {1, 2, ..., 1000} = {x | 0 < x <= 1000} • {1, 4} = {4, 1} <> {1, 4, 1} (which is not a set!) • Membership • x A - x is an element (member) of A • x A - x is not an element of A • Null Set • NULL CS 303 – Sets Lecture 8

  4. Sub-Sets • A is a SubSet of B • A is included in B • B is a SuperSet of A • A is a subset of A • NULL is a subset of A • If A is a Subset of B and B is a Subset of A then A = B • if A is a Subset of B and A <> B then A is a proper subset of B CS 303 – Sets Lecture 8

  5. (abstract) Set Operations • Union: A union B = {x | (x is in A) OR (x is in B)} • Intersection: A intersect B = {x | (x is in A) AND (x is in B)} • Difference: A-B = {x | (x is in a) AND (x is NOT in B)} • [show Venn Diagrams] CS 303 – Sets Lecture 8

  6. Set ADT - Operations • MakeNull(A): A = NULL • Member(x,A): b = (x is in A?) • Union(A,B,C): C = A UNION B • Intersection(A,B,C): C = A INTERSECT B • Difference(A,B,C): C = A – B • Equal(A,B): A == B? • Assign(A,B): A = B • Insert(x,A): A = A UNION {x} • Delete(x,A): A = A – {x} • Min(A): x = a | (a in A) • and for all z in A, a <= z • Merge(A,B): if A INTERSECT B is NOT NULL • then C = A UNION B, • else UNDEFINED! • Find(x): A = Ai | x is in Ai • if U = {Ai | i<>j • implies Ai INTERSECT Aj = NULL} CS 303 – Sets Lecture 8

  7. Specialized ADTs based on SET • UID – Union, Intersection, Difference • IDM – Insert, Delete, Member (Dictionary) • I(DeleteMin) – Insert, DeleteMin (Priority Queue) • ...and more...much more... CS 303 – Sets Lecture 8

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