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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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Sets

Model

Notation

Sub-Sets

Operations (abstract)

CS 303 – Sets

Lecture 8

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

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

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

(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

• 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