Mathematics for Computer Science
MIT 6.042J/18.062J
Partial Orders
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.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}
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.3
proper subset relation
A B
means
B has everything
that A has
and more: B A
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.4
is asymmetric
binary relation R on set A
is asymmetric:
aRb implies NOT(bRa)
for all a,b
Feb. 19, 2009
A
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.5
properties of
[A
B and B C]
implies A C
transitivity
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.6
is transitive
binary relation R on set A
is transitive:
aRb and bRc implies aRc
for all a,b,c
Feb. 19, 2009
A
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.8
strict partial orders
transitive &
asymmetric
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.9
Subject Prerequisites
subject c is a direct
prerequisite for subject d
c→d
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.11
Direct Prerequisites
18.01 → 6.042 → 6.046 → 6.840
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.12
Indirect Prerequisites
18.01 → 6.042 → 6.046 → 6.840
18.01 is indirect prerequisite
of 6.042 and 6.840
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.13
Indirect Prerequisites
18.01 → 6.042 → 6.046 → 6.840
another indirect prereq
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.14
Indirect Prerequisites
18.01 → 6.042 → 6.046 → 6.840
3 more indirect prerequisites
(→ is a special case of →)
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.15
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
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.16
Indirect Prerequisites
→ must be a strict
partial order on MIT
subjects
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.17
strict p.o. properly divides
30
10
15
3
5
2
1 on {1,2,3,5,10,15,30}
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.23
divides & subset
same "shape"
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.24
proper subset
{1,2,3,5,10,15,30}
{1,2,5,10}
{1,3,5,15}
{1,3}
{1,5}
{1,2}
{1}
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.25
p.o. has same shape as
Theorem: Every strict
partial order has the same
shape (is isomorphic to) as
some collection of subsets
partially ordered by .
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.26
Team Problems
Problems
1&2
Feb. 19, 2009
Copyright © Albert R. Meyer, 2009. All rights reserved.
lec 3R.27