Mathematics for Computer Science
MIT 6.042J/18.062J
Partial Orders
Albert R Meyer, Sep. 30, 2009
lec4W.1
proper subset relation
{1,2,3,5,10,15,30}
{1,2,5,10}
{1,3,5,15}
{1,3}
{1,5}
{1,2}
{1}
Albert R Meyer, Sep. 30, 2009
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proper subset relation
A Ã B means
B has everything
that A has
and more: B ⊄ A
Albert R Meyer, Sep. 30, 2009
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properties of
⊂
A ⊂ B implies B ⊄ A
asymmetry
Albert R Meyer, Sep. 30, 2009
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⊂ is asymmetric
binary relation R on set A
is asymmetric:
aRb implies NOT(bRa)
for all a,b A
Albert R Meyer, Sep. 30, 2009
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properties of
⊂
[A ⊂ B and B ⊂ C]
implies A ⊂ C
transitivity
Albert R Meyer, Sep. 30, 2009
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⊂ is transitive
binary relation R on set A
is transitive:
aRb and bRc implies aRc
for all a,b,c A
Albert R Meyer, Sep. 30, 2009
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strict partial orders
transitive &
asymmetric
Albert R Meyer, Sep. 30, 2009
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Subject Prerequisites
subject c is a direct
prerequisite for subject d
c→d
Albert R Meyer, Sep. 30, 2009
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Direct Prerequisites
18.01 → 6.042 → 6.046 → 6.840
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Indirect Prerequisites
18.01 → 6.042 → 6.046 → 6.840
18.01 is indirect prerequisite
of 6.042 and 6.840
Albert R Meyer, Sep. 30, 2009
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Indirect Prerequisites
18.01 → 6.042 → 6.046 → 6.840
another indirect prereq
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Indirect Prerequisites
18.01 → 6.042 → 6.046 → 6.840
3 more indirect prerequisites
(→ is a special case of →)
Albert R Meyer, Sep. 30, 2009
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Indirect Prerequisites
If subjects c, d are mutual prereq’s
c → d and d → c
then no one can graduate!
Comm. on Curricula ensures:
if c → d, then NOT(d → c)
asymmetry
Albert R Meyer, Sep. 30, 2009
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Indirect Prerequisites
→ better be a strict
partial order on MIT
subjects
Albert R Meyer, Sep. 30, 2009
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partial order: properly divides
30
10
15
3
5
2
1 on {1,2,3,5,10,15,30}
Albert R Meyer, Sep. 30, 2009
lec4W.18
same shape
as ⊂ example
Albert R Meyer, Sep. 30, 2009
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proper subset
{1,2,3,5,10,15,30}
{1,2,5,10}
{1,3,5,15}
{1,3}
{1,5}
{1,2}
{1}
Albert R Meyer, Sep. 30, 2009
lec4W.20
partial order: properly divides
30
10
15
3
5
2
1 on {1,2,3,5,10,15,30}
Albert R Meyer, Sep. 30, 2009
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same shape
as ⊂ example
isomorphic
Albert R Meyer, Sep. 30, 2009
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p.o. has same shape as
⊂
Theorem: Every strict
partial order is isomorphic
to a collection of subsets
partially ordered by ⊂.
Albert R Meyer, Sep. 30, 2009
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subsets from divides
30 {1,2,3,5,10,15,30}
15{1,3,5,15}
10 {1,2,5,10}
3 {1,3}
5 {1,5}
2 {1,2}
1 {1}
Albert R Meyer, Sep. 30, 2009
lec4W.24
p.o. has same shape as
⊂
proof: map each element, a,
to the set of elements below it
a Æ
{b ŒA | b R a
OR b = a
Albert R Meyer, Sep. 30, 2009
}
lec4W.25
weak partial orders
same as a strict partial
order R, except that
aRa always holds
reflexivity
Albert R Meyer, Sep. 30, 2009
lec4W.26
weak partial orders
same as a strict partial
order R, except that
aRa always holds
examples:
•⊆ is weak p.o. on sets
•  is weak p.o.
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Reflexivity
relation R on set A
is reflexive iff
aRa for all a A
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antisymmetr
y
binary relation R on set A
is antisymmetric iff it is
asymmetric except that
aRa is allowed to hold.
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weak partial orders
transitive
antisymmetric
& reflexive
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lec4W.33
Team Problems
Problems
1−5
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