We discussed relations and functions. A relation
R
from
A
to
B
is a subset
R
⊆
A
×
B
.
A relation on
A
×
A
can be reflexive, symmetric or transitive.
A relation which is reflexive, symmetric and transitive is called an equivalence
relation. An equivalence relation on
A
×
A
partitions
A
into equivalence classes.
A relation
R
⊆
A
×
B
is functional if
•
For each
a
∈
A
, there is an element (
a, b
)
∈
R
.
•
If (
a, b
)
∈
R
and (
a, b
0
)
∈
R
, then
b
=
b
0
.
Special kinds of functions are onetoone and onto.
If
f
:
A
→
B
is onetoone then

A
 ≤ 
B

.
If
g
:
A
→
B
is onto then

A
 ≥ 
B

.
We counted the number of functions of various kinds from a finite set
A
to
a finite set
B
.
1. Let
A
=
{
a, b, c
}
and
B
=
{
a, c, e, g
}
.
a) How many relations between
A
and
B
are there? List three of them.
b) How many relations between
B
and
A
are there? List three of them.
c) List three relations which are both from
A
to
B
and from
B
to
A
. How
many such things are there?
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 Fall '10
 PeterChristopher
 Math

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