R A B Relational Mapping b 1 Properties b 2 a 1 (Archery) a 2 b - - PDF document

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Relational Mapping Properties (Archery)

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Binary relation R from A to B

domain codomain A B

a2 a3 b1 a1 b2 b3 b4

arrows

R

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function archery ≤ 1 arrow out

F( ) =

A B

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≤, ≥ ≤, ≥

archery on relations

, = 1 arrow out

A B

≤, ≥ ≤, ≥ ,= 1 arrow in

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≥

total relation archery

1 arrow out

A B

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A B

≥

total relation archery

1 arrow out

2

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A B

≥

total relation archery

1 arrow out

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total relation

R is total iff

A = R−1 B

( )

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total & function archery

exactly 1 arrow out

A B

F( ) =

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1 g(x, y) ::= x − y

domain(g) = all pairs of reals codomain(g) = all reals But g is not total: g(r,r) not defined

g : R×R → R

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1 g0(x, y) ::= x − y

where g0, g have the same graph, different domains

g0 is total

g 0 :D → R

D ::= R2 − (x, y)|x = y

{ }

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≥

surjection archery

1 arrow in

A B

3

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≥

surjection archery

1 arrow in

A B

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≥

surjection archery

1 arrow in

A B

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surjection

R is a surjection iff

R A ( ) = B

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A B

injection archery

≤ 1 arrow in

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A B

injection archery

≤ 1 arrow in

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A B

injection archery

≤ 1 arrow in

4

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B A

f( ) = bijection archery

exactly 1 arrow out exactly 1 arrow in

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Mapping Rule (bij)

A bijection from A to B implies

|A| = |B| for finite A, B

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6.042J / 18.062J Mathematics for Computer Science

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