# Discrete Structure

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```					Discrete Structure
Li Tak Sing(李德成)
Equivalence relations

Any binary relation that is reflexive,
symmetric, and transitive is called an
equivalence relation.
Sample equivalence relations

R is over N, xRy means x+y is even
R is over real number, xRy means
sin(x)=sin(y)
Examples

Let R be defined on N by xRy iff |x-y| is
odd. Show that R is not an equivalance
relation on N.
Given the relation over the integers
defined by a~b iff |a|=|b|, either prove that
~ is an equivalence relation or prove that ~
not an equivalence relation.
Intersection property of equivalence

If E and F are equivalence relations on the
set A, then EF is an equivalence relation
on A.
Kernel relations

If f is a function with domain A, then the
relation ~ defined by
x~y iff f(x)=f(y)
is an equivalence relation on A, and it is
called the kernel relation of f.
Equivalence classes

Let R be an equivalence relation on a set
S. If aS, then the equivalence class of a,
denoted by [a], is the subset of S
consisting of all elements that are
equivalent to a. In other words, we have
[a]={x S | xRa}
Example of equivalence class

a~b iff a+b is even
a~b iff sin(a)=sin(b)
a~b iff a and b are studying the same
course
Property of equivalences
 Let S be a set with an equivalence relation R. If
a, b S, then either [a]=[b] or [a][b]=.
 Proof. If [a][b], and [a][b]  , there is an
element c so that c[a] and c[b]. So we have
aRc and cRb, therefore aRb. Now, for any x
[a] , xRa. Since aRb, therefore we have xRb
which implies that x [b]. Therefore, [a][b].
Similarly, we have [b][a].
Partitions
 By a partition of a set we mean a collection of
nonempty subsets that are disjoint from each
other and whose union is the whole set.
 If R is an equivalence relation on the set S, then
the equivalence classes form a partition of S.
Conversely, if P is a partition of a set S, then
there is an equivalence relation on S whose
equivalence classes are sets of P.
Refinement of a partition

Suppose that P and Q are two partitions of
a set S. If each set of P is a subset of a
set in Q, then P is a refinement of Q. The
finest of all partitions on S is the collection
of singleton sets. The coarsest of all
partitions of S is the set S itself.
Example

aRb iff ab (mod 2)
aSb iff ab (mod 4)
Example

Given the following set of words.
{rot, tot, root, toot, roto, toto, too, to otto}.
Let f be the funtion that maps word to its
set of letters. For kernel relation of f,
describe the equivalence classes.
Let f be the function that maps a word to
its bag of letters. For the kernel relation of
f, describe the equivalence classes.
Kruskai's algorithm for minimal
spanning trees
In the spanning tree problems, we can
define a relation R so that aRb if there is a
path between a and b. The algorithm is:
Kruskal's Algorithm
 Sort the edges of the graph by weight, and let L be the
sorted list.
 Let T be the minimal spanning tree and initialize T:=.
 For each vertex v of the graph, create the equivalence
class [v]={v}
 while there are 2 or more equivalence classes do
Let {a,b} be the edge at the head of L;
L:=tail(L);
if [a][b] then
T:=T{{a,b}};
Replace the equivalence classes [a] and [b] by [a]
[b]
fi
 od
Spanning trees
b
a           2

1
2
1                   c
2       d       2
e
1
2
f
Spanning trees
a

2
1
e
3                   2

2
b   1 f
3
d
1                   1

c

```
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