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Lect 9 - October 23, 2008 - ECS 20 - Fall 2008 - Phil Rogaway
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Today: o Quiz 2
o Relations
o Functions
- SCRIBE (none today, use your own notes for the class notes)
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1. Review of relations
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DEF: A and B sets. Then a *relation* R is subset of A x B.
R \subseteq A x B
Defined last time:
inverse of a relation
composition of a relation
Looked at: a graphical interpretation of relations
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2. Properties of Relations
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Reflexive: x R x for all x
Symmetric: x R y -> y R x for all x,y
Transitive: x R y and y R x -> x R z for all x, y, z
if R has all three properties, R is said to be an equivalence relation
Make a table....
Reflexive Symmetric Transitive comments
= on Integers Yes Yes Yes
(or anything else)
<= , integers Yes No Yes actually _antisymmetric_
Define this
subseteq, sets Yes No Yes antisymmetric
x E y if x and y are
regular expressions and Yes Yes Yes blocks are the regular
L(x) = L(y) languages
x S y if x is a substring Yes No Yes
of y
x R y where x and y are
strings and M is a some
DFA and you go to the Yes Yes Yes
same state on processing
x and y
x | y if 3 | x-y Yes Yes Yes Carefully prove this one
and write out its blocks.
Define when n | m
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3. Equivalence classes, quotients
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Important definition:
If R is an equivalence relation on A x A then [x] denotes the set
of all elements related to x:
[x] = { x': x R x'}
We call [x] the *equivalence class* (or *block*) of x.
The set of all equivalence classes of A with respect to a
relation R is denoted A/R "quotient set of A by R", or "A mod R".
I claim that ever equivalence relation
*partitions* the universe into its blocks?
What does this mean?
Define a partitioning of the set A:
Def: {A_i: i in I} is a partition of A if each A_i is nonempty,
their union is A, and they are pairwise disjoint.
Proposition: Let R be an equivalence relation on a set A.
Then the blocks of R are a partition of A.
Proof: didn't get to it.
NOTATION: A/R the blocks of A relative to equivalence relation R.
Note: you can talk about the blocks being related
to one another by R, [x] R [y] iff x R y.
Well-defined.
Now go back to prior examples and identify the blocks in each case.
When introduced partition, a student asked a question of we had
earlier used our *tiles* to partition the (upper right quadrant) of the
plane. I indicated that we had, but that one needs to be careful at the
"edges" of each tile to make sure that each point is in only one tile.
We defined
[a, b) = {x\in\R a <= x < b)
So a tile with left endpoint at (i,j) is is [i, i+1) x [j, j+1)
and the plane is the disjoint union of tiles
T_ij = [i, i+1) x [j, j+1)
when i,j \in \N