<s>
A	O
Venn	B-Application
diagram	I-Application
is	O
a	O
widely	O
used	O
diagram	O
style	O
that	O
shows	O
the	O
logical	O
relation	O
between	O
sets	O
,	O
popularized	O
by	O
John	O
Venn	O
(	O
1834	O
–	O
1923	O
)	O
in	O
the	O
1880s	O
.	O
</s>
<s>
The	O
diagrams	O
are	O
used	O
to	O
teach	O
elementary	O
set	O
theory	O
,	O
and	O
to	O
illustrate	O
simple	O
set	O
relationships	O
in	O
probability	O
,	O
logic	O
,	O
statistics	O
,	O
linguistics	O
and	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
A	O
Venn	B-Application
diagram	I-Application
uses	O
simple	O
closed	O
curves	O
drawn	O
on	O
a	O
plane	O
to	O
represent	O
sets	O
.	O
</s>
<s>
A	O
Venn	B-Application
diagram	I-Application
may	O
also	O
be	O
called	O
a	O
set	B-Application
diagram	I-Application
or	O
logic	B-Application
diagram	I-Application
.	O
</s>
<s>
A	O
Venn	B-Application
diagram	I-Application
consists	O
of	O
multiple	O
overlapping	O
closed	O
curves	O
,	O
usually	O
circles	O
,	O
each	O
representing	O
a	O
set	O
.	O
</s>
<s>
In	O
Venn	B-Application
diagrams	I-Application
,	O
the	O
curves	O
are	O
overlapped	O
in	O
every	O
possible	O
way	O
,	O
showing	O
all	O
possible	O
relations	O
between	O
the	O
sets	O
.	O
</s>
<s>
They	O
are	O
thus	O
a	O
special	O
case	O
of	O
Euler	B-Application
diagrams	I-Application
,	O
which	O
do	O
not	O
necessarily	O
show	O
all	O
relations	O
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
were	O
conceived	O
around	O
1880	O
by	O
John	O
Venn	O
.	O
</s>
<s>
They	O
are	O
used	O
to	O
teach	O
elementary	O
set	O
theory	O
,	O
as	O
well	O
as	O
illustrate	O
simple	O
set	O
relationships	O
in	O
probability	O
,	O
logic	O
,	O
statistics	O
,	O
linguistics	O
,	O
and	O
computer	B-General_Concept
science	I-General_Concept
.	O
</s>
<s>
A	O
Venn	B-Application
diagram	I-Application
in	O
which	O
the	O
area	O
of	O
each	O
shape	O
is	O
proportional	O
to	O
the	O
number	O
of	O
elements	O
it	O
contains	O
is	O
called	O
an	O
area-proportional	O
(	O
or	O
scaled	O
)	O
Venn	B-Application
diagram	I-Application
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
were	O
introduced	O
in	O
1880	O
by	O
John	O
Venn	O
in	O
a	O
paper	O
entitled	O
"	O
On	O
the	O
Diagrammatic	O
and	O
Mechanical	O
Representation	O
of	O
Propositions	O
and	O
Reasonings	O
"	O
in	O
the	O
Philosophical	O
Magazine	O
and	O
Journal	O
of	O
Science	O
,	O
about	O
the	O
different	O
ways	O
to	O
represent	O
propositions	O
by	O
diagrams	O
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
are	O
very	O
similar	O
to	O
Euler	B-Application
diagrams	I-Application
,	O
which	O
were	O
invented	O
by	O
Leonhard	O
Euler	O
in	O
the	O
18th	O
century	O
.	O
</s>
<s>
Venn	O
did	O
not	O
use	O
the	O
term	O
"	O
Venn	B-Application
diagram	I-Application
"	O
and	O
referred	O
to	O
the	O
concept	O
as	O
"	O
Eulerian	B-Application
Circles	I-Application
"	O
.	O
</s>
<s>
He	O
became	O
acquainted	O
with	O
Euler	B-Application
diagrams	I-Application
in	O
1862	O
and	O
wrote	O
that	O
Venn	B-Application
diagrams	I-Application
did	O
not	O
occur	O
to	O
him	O
"	O
till	O
much	O
later	O
"	O
,	O
while	O
attempting	O
to	O
adapt	O
Euler	B-Application
diagrams	I-Application
to	O
Boolean	O
logic	O
.	O
</s>
<s>
In	O
the	O
opening	O
sentence	O
of	O
his	O
1880	O
article	O
Venn	O
wrote	O
that	O
Euler	B-Application
diagrams	I-Application
were	O
the	O
only	O
diagrammatic	O
representation	O
of	O
logic	O
to	O
gain	O
"	O
any	O
general	O
acceptance	O
"	O
.	O
</s>
<s>
The	O
term	O
"	O
Venn	B-Application
diagram	I-Application
"	O
was	O
later	O
used	O
by	O
Clarence	O
Irving	O
Lewis	O
in	O
1918	O
,	O
in	O
his	O
book	O
A	O
Survey	O
of	O
Symbolic	O
Logic	O
.	O
</s>
<s>
In	O
the	O
20th	O
century	O
,	O
Venn	B-Application
diagrams	I-Application
were	O
further	O
developed	O
.	O
</s>
<s>
He	O
also	O
showed	O
that	O
such	O
symmetric	O
Venn	B-Application
diagrams	I-Application
exist	O
when	O
n	O
is	O
five	O
or	O
seven	O
.	O
</s>
<s>
In	O
2002	O
,	O
Peter	O
Hamburger	O
found	O
symmetric	O
Venn	B-Application
diagrams	I-Application
for	O
n	O
=	O
11	O
and	O
in	O
2003	O
,	O
Griggs	O
,	O
Killian	O
,	O
and	O
Savage	O
showed	O
that	O
symmetric	O
Venn	B-Application
diagrams	I-Application
exist	O
for	O
all	O
other	O
primes	O
.	O
</s>
<s>
These	O
combined	O
results	O
show	O
that	O
rotationally	O
symmetric	O
Venn	B-Application
diagrams	I-Application
exist	O
,	O
if	O
and	O
only	O
if	O
n	O
is	O
a	O
prime	O
number	O
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
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
,	O
as	O
part	O
of	O
the	O
new	O
math	O
movement	O
in	O
the	O
1960s	O
.	O
</s>
<s>
A	O
Venn	B-Application
diagram	I-Application
is	O
constructed	O
with	O
a	O
collection	O
of	O
simple	O
closed	O
curves	O
drawn	O
in	O
a	O
plane	O
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
normally	O
comprise	O
overlapping	O
circles	O
.	O
</s>
<s>
For	O
instance	O
,	O
in	O
a	O
two-set	O
Venn	B-Application
diagram	I-Application
,	O
one	O
circle	O
may	O
represent	O
the	O
group	O
of	O
all	O
wooden	O
objects	O
,	O
while	O
the	O
other	O
circle	O
may	O
represent	O
the	O
set	O
of	O
all	O
tables	O
.	O
</s>
<s>
Shapes	O
other	O
than	O
circles	O
can	O
be	O
employed	O
as	O
shown	O
below	O
by	O
Venn	O
's	O
own	O
higher	O
set	B-Application
diagrams	I-Application
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
do	O
not	O
generally	O
contain	O
information	O
on	O
the	O
relative	O
or	O
absolute	O
sizes	O
(	O
cardinality	B-Application
)	O
of	O
sets	O
.	O
</s>
<s>
That	O
is	O
,	O
they	O
are	O
schematic	B-Application
diagrams	I-Application
generally	O
not	O
drawn	O
to	O
scale	O
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
are	O
similar	O
to	O
Euler	B-Application
diagrams	I-Application
.	O
</s>
<s>
However	O
,	O
a	O
Venn	B-Application
diagram	I-Application
for	O
n	O
component	O
sets	O
must	O
contain	O
all	O
2n	O
hypothetically	O
possible	O
zones	O
,	O
that	O
correspond	O
to	O
some	O
combination	O
of	O
inclusion	O
or	O
exclusion	O
in	O
each	O
of	O
the	O
component	O
sets	O
.	O
</s>
<s>
Euler	B-Application
diagrams	I-Application
contain	O
only	O
the	O
actually	O
possible	O
zones	O
in	O
a	O
given	O
context	O
.	O
</s>
<s>
In	O
Venn	B-Application
diagrams	I-Application
,	O
a	O
shaded	O
zone	O
may	O
represent	O
an	O
empty	O
zone	O
,	O
whereas	O
in	O
an	O
Euler	B-Application
diagram	I-Application
,	O
the	O
corresponding	O
zone	O
is	O
missing	O
from	O
the	O
diagram	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
one	O
set	O
represents	O
dairy	O
products	O
and	O
another	O
cheeses	O
,	O
the	O
Venn	B-Application
diagram	I-Application
contains	O
a	O
zone	O
for	O
cheeses	O
that	O
are	O
not	O
dairy	O
products	O
.	O
</s>
<s>
Assuming	O
that	O
in	O
the	O
context	O
cheese	O
means	O
some	O
type	O
of	O
dairy	O
product	O
,	O
the	O
Euler	B-Application
diagram	I-Application
has	O
the	O
cheese	O
zone	O
entirely	O
contained	O
within	O
the	O
dairy-product	O
zone	O
—	O
there	O
is	O
no	O
zone	O
for	O
(	O
non-existent	O
)	O
non-dairy	O
cheese	O
.	O
</s>
<s>
This	O
means	O
that	O
as	O
the	O
number	O
of	O
contours	O
increases	O
,	O
Euler	B-Application
diagrams	I-Application
are	O
typically	O
less	O
visually	O
complex	O
than	O
the	O
equivalent	O
Venn	B-Application
diagram	I-Application
,	O
particularly	O
if	O
the	O
number	O
of	O
non-empty	O
intersections	O
is	O
small	O
.	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
diagram	I-Application
of	O
those	O
sets	O
are	O
:	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
typically	O
represent	O
two	O
or	O
three	O
sets	O
,	O
but	O
there	O
are	O
forms	O
that	O
allow	O
for	O
higher	O
numbers	O
.	O
</s>
<s>
Shown	O
below	O
,	O
four	O
intersecting	O
spheres	O
form	O
the	O
highest	O
order	O
Venn	B-Application
diagram	I-Application
that	O
has	O
the	O
symmetry	O
of	O
a	O
simplex	O
and	O
can	O
be	O
visually	O
represented	O
.	O
</s>
<s>
The	O
16	O
intersections	O
correspond	O
to	O
the	O
vertices	O
of	O
a	O
tesseract	B-Language
(	O
or	O
the	O
cells	O
of	O
a	O
16-cell	O
,	O
respectively	O
)	O
.	O
</s>
<s>
He	O
also	O
gave	O
a	O
construction	O
for	O
Venn	B-Application
diagrams	I-Application
for	O
any	O
number	O
of	O
sets	O
,	O
where	O
each	O
successive	O
curve	O
that	O
delimits	O
a	O
set	O
interleaves	O
with	O
previous	O
curves	O
,	O
starting	O
with	O
the	O
three-circle	O
diagram	O
.	O
</s>
<s>
Anthony	O
William	O
Fairbank	O
Edwards	O
constructed	O
a	O
series	O
of	O
Venn	B-Application
diagrams	I-Application
for	O
higher	O
numbers	O
of	O
sets	O
by	O
segmenting	O
the	O
surface	O
of	O
a	O
sphere	O
,	O
which	O
became	O
known	O
as	O
Edwards	O
–	O
Venn	B-Application
diagrams	I-Application
.	O
</s>
<s>
Edwards	O
–	O
Venn	B-Application
diagrams	I-Application
are	O
topologically	O
equivalent	O
to	O
diagrams	O
devised	O
by	O
Branko	O
Grünbaum	O
,	O
which	O
were	O
based	O
around	O
intersecting	O
polygons	B-General_Concept
with	O
increasing	O
numbers	O
of	O
sides	O
.	O
</s>
<s>
They	O
are	O
also	O
two-dimensional	O
representations	O
of	O
hypercubes	B-Operating_System
.	O
</s>
<s>
Joaquin	O
and	O
Boyles	O
,	O
on	O
the	O
other	O
hand	O
,	O
proposed	O
supplemental	O
rules	O
for	O
the	O
standard	O
Venn	B-Application
diagram	I-Application
,	O
in	O
order	O
to	O
account	O
for	O
certain	O
problem	O
cases	O
.	O
</s>
<s>
For	O
instance	O
,	O
regarding	O
the	O
issue	O
of	O
representing	O
singular	O
statements	O
,	O
they	O
suggest	O
to	O
consider	O
the	O
Venn	B-Application
diagram	I-Application
circle	O
as	O
a	O
representation	O
of	O
a	O
set	O
of	O
things	O
,	O
and	O
use	O
first-order	O
logic	O
and	O
set	O
theory	O
to	O
treat	O
categorical	O
statements	O
as	O
statements	O
about	O
sets	O
.	O
</s>
<s>
So	O
,	O
for	O
example	O
,	O
to	O
represent	O
the	O
statement	O
"	O
a	O
is	O
F	O
"	O
in	O
this	O
retooled	O
Venn	B-Application
diagram	I-Application
,	O
a	O
small	O
letter	O
"	O
a	O
"	O
may	O
be	O
placed	O
inside	O
the	O
circle	O
that	O
represents	O
the	O
set	O
F	O
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
correspond	O
to	O
truth	O
tables	O
for	O
the	O
propositions	O
,	O
,	O
etc.	O
,	O
in	O
the	O
sense	O
that	O
each	O
region	O
of	O
Venn	B-Application
diagram	I-Application
corresponds	O
to	O
one	O
row	O
of	O
the	O
truth	O
table	O
.	O
</s>
<s>
Another	O
way	O
of	O
representing	O
sets	O
is	O
with	O
John	O
F	O
.	O
Randolph	O
's	O
R-diagrams	B-Application
.	O
</s>
