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Set Theory

Foundation of Relational Database Systems

E.F. Codd

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Basic Concepts

A set is a collection of elements

From a known background “Universe”

A definition we are satisfied with

Everything is in the “Universe”

An element could be any thing

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Examples

• All UWP students in CS363 this semester

A = {UWP students in CS363 this semester}

• B = {All UWP students who play Bridge}

• C = {1, 2, 3, 4}

• I = {i: i is an integer}

• Number of elements of each set

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No repeating elements

{1, 2, 3, 4, 2}

Not a set in classic set theory

Elements are not ordered

{1, 2, 3, 4}

{2, 4, 1, 3}

The same set

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Empty Set

A = {UWP students in CS363 this semester}

D = {s | s in A and has attended Turing

Award ceremony}

D = {}

=

This is not empty set: {}

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Sets and Elements

S is a set

X is an element in the “Universe”

Two possibilities:

x is in S

(x S)

or

x is not in S

(x S)

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Subsets

For two sets A and B and any element x,if x A then x B

Then A is a subset of BA B or B A

(similar to A < B or B > A)

A could be the same as B A B or B A (similar to A <= B or B >= A)

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Subsets

For two sets A and B and any element x,if x A then x B

Then A is a subset of B

A is not a subset of BThere is an element x such that

x A and x B

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Subsets (II)

For any set X,

X

No element e such that

e and e X

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Proper Subsets

For any set X, X X X

A is a proper subset of B, if all the three conditions are true:

A B (A B) A B A

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Cardinality

C = {1, 2, 3, 4} |C| = 4 A = {UWP students in CS363 this semester} |A| = 55

I = {i: i is an integer} |I| =

(Number of elements of finite sets)

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Power Set

For any set X, P(X) = {S | S is a subset of X} = {S | S X}

C = {1, 2, 3, 4}

P(C) = {{1}, {2}, {3}, {4},

{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},

{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4},

{1, 2, 3, 4},

{}}

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Power Set

C = {1, 2, 3, 4}

P(C) = {{1}, {2}, {3}, {4},

{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},

{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4},

{1, 2, 3, 4},

{}}

P(C) = {{1}, {2}, {3}, {4},

{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},

{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4},

{1, 2, 3, 4},

}

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The cardinality of the power set

X is a set and its cardinality is |X|

The power set of X is P(X) and its cardinality is |P(X)| = 2|X|

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Example I

C = {1, 2, 3, 4}|C| = 4

P(C) = {{1}, {2}, {3}, {4},

{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4},

{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4},

{1, 2, 3, 4},

}

|P(C)| = 24 = 16

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Example II

X = {x0, x1, x2, x3, x4, x5, x6, x7} |X| = 8 |P(X)| = 28

x0 x1 x2 x3 x4 x5 x6 x7 1 0 0 1 1 0 0 0

The same as a byte with 8 bits.

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Set Operations

Set UnionA B = {x: x A or x B}

A = {1, 2, 3, 4}

B = {2, 5}

A B = {x: x A or x B}

= {1, 2, 3, 4, 5}

= B A

Range of |A B|?

Max (|A|, |B|) <= |A B| <= |A| + |B|

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Set IntersectionA B = {x: x A and x B}

A = {1, 2, 3, 4}B = {2, 5}A B = {x: x A and x B}

= {2} = B A

Range of |A B| ?0 <= |A B| <= Min(|A|, |B|)

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Set Difference

A – B = {x: x A but x B}

A = {1, 2, 3, 4}B = {2, 5}

A – B = {x: x A but x B} = {1, 3, 4}

B - AB – A = {5}

Range of |A – B|?

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Cartesian Product

A B = {(a, b): a A and b B}

A = {1, 2, 3, 4} B = {2, 5}A B = {(a, b): a A and b B}

= {(1, 2), (1, 5), (2, 2), (2, 5), (3, 2), (3, 5), (4, 2), (4, 5)}

A B B A

|A B| = |A| |B|

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Cartesian Product (II) Multiple sets A1, A2, A3, …, AnA1 A2 A3 … An = {(x1, x2, x3, …, xn): xi Ai}

It is possible for some i and j, Ai = Aj.

A = {1, 2, 3, 4}B = {2, 5}A B B = {(a, b, c): a A and b B and c B}

= {(1, 2, 2), (1, 5, 2), (2, 2, 2), (2, 5, 2), (3, 2, 2), (3, 5, 2), (4, 2, 2), (4, 5, 2),

(1, 2, 5), (1, 5, 5), (2, 2, 5), (2, 5, 5), (3, 2, 5), (3, 5, 5), (4, 2, 5), (4, 5, 5)}

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Assignment 1

Due Wednesday, January 30At beginning of classType your work and submit a hard copy.