Mathematics for Computer Science

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Mathematics for Computer Science

• Lecture 6 Relations
• Leon van der Torre

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Slides

• Slides can be found on the internet
• These slides are based on MIT open courseware,
  by A. Meyer

BECS - Bachelor of Engineering in Computer
Science
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Binary relation R from A to B
codomain
domain
graph(R)
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Example
Classes
Students
6.042 6.003 6.012
is taking
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Example
Arithmetic Expressions
values
12 Sqrt(9) 50/10 - 3
3 5 2
evaluates to
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Example
Cities
Cities
Boston Providence New York
Boston Providence New York
direct bus connection
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Example
direct bus connection
Cities
Boston Providence New York
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Relation Abstraction
(Binary) Relation domain set A codomain
set B graph subset of A ? B
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Asymmetry
Binary relation, R, on set A, is asymmetric iff
aRb implies ?(bRa) for all a,b ? A
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Transitivity
Binary relation, R, on set A, is transitive
aRb and bRc implies aRc for all a,b,c ? A.
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Strict Partial Orders

• Binary relation, R, on set A,
• is a strict partial order iff
• it is transitive and
• asymmetric

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Irreflexivity

• If R is a strict partial order, then
• ?(aRa)
• for all a? A

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Weak Partial Orders
same as strict except aRa for all a ?
A (reflexivity)
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Ordering Relations

• on the Integers
• lt on the Reals
• ? on Sets (subset)
• ? on Sets (proper subset)

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Partial Orders

• y ltlt x (much less than)
• (say, y 2 ? x)
• ? 3 ltlt 4

? 4 ltlt 3
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Representing Partial Orders

• The subset relation, ?
• on sets is the canonical
• example of weak partial order

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(Proper) Subset Relation
1
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Partial Order divides

• a divides b (a b) iff
• ka b for some k??

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Partial Order divides
210
2
12
1
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Subset Relation
1
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Divides Subset

• same "shape"

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Subset Relation
15?1,3,5,15
3 ?1,3
1 ?1
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Properties of ?
A ? B and B ? C implies A ? C Transitive
A ? B implies ?(B ? A) for A ? B Antisymmetric

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A/Antisymmetry

• minor difference
• whether aRa is allowed

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Total Order on A

• Partial Order, R, such that

aRb or bRa for all a?b ?A
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Strict Total Order

• a lt b or b lt a
• (for numbers a ? b)

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Weak Total Order

• a ? b or b ? a
• (for all a, b)

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Homework

• Outils mathématiques pour l'informaticien
• Section 3.1,3.4,3.6
• Section 5.1