<s>
An	O
Euler	B-Application
diagram	I-Application
(	O
,	O
)	O
is	O
a	O
diagrammatic	O
means	O
of	O
representing	O
sets	O
and	O
their	O
relationships	O
.	O
</s>
<s>
They	O
are	O
similar	O
to	O
another	O
set	O
diagramming	O
technique	O
,	O
Venn	B-Application
diagrams	I-Application
.	O
</s>
<s>
Unlike	O
Venn	B-Application
diagrams	I-Application
,	O
which	O
show	O
all	O
possible	O
relations	O
between	O
different	O
sets	O
,	O
the	O
Euler	B-Application
diagram	I-Application
shows	O
only	O
relevant	O
relationships	O
.	O
</s>
<s>
The	O
first	O
use	O
of	O
"	O
Eulerian	B-Application
circles	I-Application
"	O
is	O
commonly	O
attributed	O
to	O
Swiss	O
mathematician	O
Leonhard	O
Euler	O
(	O
1707	O
–	O
1783	O
)	O
.	O
</s>
<s>
In	O
the	O
United	O
States	O
,	O
both	O
Venn	O
and	O
Euler	B-Application
diagrams	I-Application
were	O
incorporated	O
as	O
part	O
of	O
instruction	O
in	O
set	O
theory	O
as	O
part	O
of	O
the	O
new	O
math	O
movement	O
of	O
the	O
1960s	O
.	O
</s>
<s>
Euler	B-Application
diagrams	I-Application
consist	O
of	O
simple	O
closed	O
shapes	O
in	O
a	O
two-dimensional	O
plane	O
that	O
each	O
depict	O
a	O
set	O
or	O
category	O
.	O
</s>
<s>
Curves	O
that	O
do	O
not	O
overlap	O
represent	O
disjoint	B-Algorithm
sets	I-Algorithm
,	O
which	O
have	O
no	O
elements	O
in	O
common	O
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
are	O
a	O
more	O
restrictive	O
form	O
of	O
Euler	B-Application
diagrams	I-Application
.	O
</s>
<s>
A	O
Venn	B-Application
diagram	I-Application
must	O
contain	O
all	O
2n	O
logically	O
possible	O
zones	O
of	O
overlap	O
between	O
its	O
n	O
curves	O
,	O
representing	O
all	O
combinations	O
of	O
inclusion/exclusion	O
of	O
its	O
constituent	O
sets	O
.	O
</s>
<s>
Regions	O
not	O
part	O
of	O
the	O
set	O
are	O
indicated	O
by	O
coloring	O
them	O
black	O
,	O
in	O
contrast	O
to	O
Euler	B-Application
diagrams	I-Application
,	O
where	O
membership	O
in	O
the	O
set	O
is	O
indicated	O
by	O
overlap	O
as	O
well	O
as	O
color	O
.	O
</s>
<s>
In	O
his	O
1881	O
Symbolic	O
Logic	O
Chapter	O
V	O
"	O
Diagrammatic	O
Representation	O
"	O
,	O
John	O
Venn	O
(	O
1834	O
–	O
1923	O
)	O
comments	O
on	O
the	O
remarkable	O
prevalence	O
of	O
the	O
Euler	B-Application
diagram	I-Application
:	O
</s>
<s>
Whatever	O
the	O
case	O
,	O
armed	O
with	O
these	O
observations	O
and	O
criticisms	O
,	O
Venn	O
then	O
demonstrates	O
(	O
pp.100	O
–	O
125	O
)	O
how	O
he	O
derived	O
what	O
has	O
become	O
known	O
as	O
his	O
Venn	B-Application
diagrams	I-Application
from	O
the	O
"	O
old	O
...	O
Euler	O
diagrams.	O
"	O
</s>
<s>
In	O
modern	O
usage	O
the	O
Venn	B-Application
diagram	I-Application
includes	O
a	O
"	O
box	O
"	O
that	O
surrounds	O
all	O
the	O
circles	O
;	O
this	O
is	O
called	O
the	O
universe	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
or	O
the	O
domain	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
in	O
chapter	O
"	O
Boolean	O
Algebra	O
"	O
,	O
Hill	O
and	O
Peterson	O
(	O
1968	O
,	O
1964	O
)	O
present	O
sections	O
4.5ff	O
"	O
Set	O
Theory	O
as	O
an	O
Example	O
of	O
Boolean	O
Algebra	O
"	O
,	O
and	O
in	O
it	O
they	O
present	O
the	O
Venn	B-Application
diagram	I-Application
with	O
shading	O
and	O
all	O
.	O
</s>
<s>
They	O
give	O
examples	O
of	O
Venn	B-Application
diagrams	I-Application
to	O
solve	O
example	O
switching-circuit	O
problems	O
,	O
but	O
end	O
up	O
with	O
this	O
statement	O
:	O
</s>
<s>
"	O
For	O
more	O
than	O
three	O
variables	O
,	O
the	O
basic	O
illustrative	O
form	O
of	O
the	O
Venn	B-Application
diagram	I-Application
is	O
inadequate	O
.	O
</s>
<s>
In	O
Veitch	O
's	O
method	O
the	O
variables	O
are	O
arranged	O
in	O
a	O
rectangle	O
or	O
square	O
;	O
as	O
described	O
in	O
Karnaugh	O
map	O
,	O
Karnaugh	O
in	O
his	O
method	O
changed	O
the	O
order	O
of	O
the	O
variables	O
to	O
correspond	O
to	O
what	O
has	O
become	O
known	O
as	O
(	O
the	O
vertices	O
of	O
)	O
a	O
hypercube	B-Operating_System
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
are	O
a	O
more	O
restrictive	O
form	O
of	O
Euler	B-Application
diagrams	I-Application
.	O
</s>
<s>
A	O
Venn	B-Application
diagram	I-Application
must	O
contain	O
all	O
2n	O
logically	O
possible	O
zones	O
of	O
overlap	O
between	O
its	O
n	O
curves	O
,	O
representing	O
all	O
combinations	O
of	O
inclusion/exclusion	O
of	O
its	O
constituent	O
sets	O
.	O
</s>
<s>
Regions	O
not	O
part	O
of	O
the	O
set	O
are	O
indicated	O
by	O
coloring	O
them	O
black	O
,	O
in	O
contrast	O
to	O
Euler	B-Application
diagrams	I-Application
,	O
where	O
membership	O
in	O
the	O
set	O
is	O
indicated	O
by	O
overlap	O
as	O
well	O
as	O
color	O
.	O
</s>
<s>
When	O
the	O
number	O
of	O
sets	O
grows	O
beyond	O
3	O
a	O
Venn	B-Application
diagram	I-Application
becomes	O
visually	O
complex	O
,	O
especially	O
compared	O
to	O
the	O
corresponding	O
Euler	B-Application
diagram	I-Application
.	O
</s>
<s>
The	O
difference	O
between	O
Euler	O
and	O
Venn	B-Application
diagrams	I-Application
can	O
be	O
seen	O
in	O
the	O
following	O
example	O
.	O
</s>
<s>
The	O
Euler	O
and	O
the	O
Venn	B-Application
diagrams	I-Application
of	O
those	O
sets	O
are	O
:	O
</s>
<s>
In	O
a	O
logical	O
setting	O
,	O
one	O
can	O
use	O
model-theoretic	O
semantics	O
to	O
interpret	O
Euler	B-Application
diagrams	I-Application
,	O
within	O
a	O
universe	B-Algorithm
of	I-Algorithm
discourse	I-Algorithm
.	O
</s>
<s>
In	O
the	O
examples	O
below	O
,	O
the	O
Euler	B-Application
diagram	I-Application
depicts	O
that	O
the	O
sets	O
Animal	O
and	O
Mineral	O
are	O
disjoint	O
since	O
the	O
corresponding	O
curves	O
are	O
disjoint	O
,	O
and	O
also	O
that	O
the	O
set	O
Four	O
Legs	O
is	O
a	O
subset	O
of	O
the	O
set	O
of	O
Animals	O
.	O
</s>
<s>
The	O
Venn	B-Application
diagram	I-Application
,	O
which	O
uses	O
the	O
same	O
categories	O
of	O
Animal	O
,	O
Mineral	O
,	O
and	O
Four	O
Legs	O
,	O
does	O
not	O
encapsulate	O
these	O
relationships	O
.	O
</s>
<s>
Traditionally	O
the	O
emptiness	O
of	O
a	O
set	O
in	O
Venn	B-Application
diagrams	I-Application
is	O
depicted	O
by	O
shading	O
in	O
the	O
region	O
.	O
</s>
<s>
Euler	B-Application
diagrams	I-Application
represent	O
emptiness	O
either	O
by	O
shading	O
or	O
by	O
the	O
absence	O
of	O
a	O
region	O
.	O
</s>
<s>
In	O
the	O
adjacent	O
diagram	O
,	O
examples	O
of	O
small	O
Venn	B-Application
diagrams	I-Application
are	O
transformed	O
into	O
Euler	B-Application
diagrams	I-Application
by	O
sequences	O
of	O
transformations	O
;	O
some	O
of	O
the	O
intermediate	O
diagrams	O
have	O
concurrency	O
of	O
curves	O
.	O
</s>
<s>
However	O
,	O
this	O
sort	O
of	O
transformation	O
of	O
a	O
Venn	B-Application
diagram	I-Application
with	O
shading	O
into	O
an	O
Euler	B-Application
diagram	I-Application
without	O
shading	O
is	O
not	O
always	O
possible	O
.	O
</s>
<s>
There	O
are	O
examples	O
of	O
Euler	B-Application
diagrams	I-Application
with	O
9	O
sets	O
that	O
are	O
not	O
drawable	O
using	O
simple	O
closed	O
curves	O
without	O
the	O
creation	O
of	O
unwanted	O
zones	O
since	O
they	O
would	O
have	O
to	O
have	O
non-planar	O
dual	O
graphs	O
.	O
</s>
<s>
This	O
example	O
shows	O
the	O
Euler	O
and	O
Venn	B-Application
diagrams	I-Application
and	O
Karnaugh	O
map	O
deriving	O
and	O
verifying	O
the	O
deduction	O
"	O
No	O
Xs	O
are	O
Zs	O
"	O
.	O
</s>
<s>
Given	O
the	O
example	O
above	O
,	O
the	O
formula	O
for	O
the	O
Euler	O
and	O
Venn	B-Application
diagrams	I-Application
is	O
:	O
</s>
