<s>
A	O
Randolph	B-Application
diagram	I-Application
(	O
R-diagram	B-Application
)	O
is	O
a	O
simple	O
way	O
to	O
visualize	O
logical	O
expressions	O
and	O
combinations	O
of	O
sets	O
.	O
</s>
<s>
Randolph	B-Application
diagrams	I-Application
were	O
created	O
by	O
mathematician	O
John	O
F	O
.	O
Randolph	O
in	O
1965	O
,	O
while	O
he	O
was	O
teaching	O
at	O
the	O
University	O
of	O
Arkansas	O
.	O
</s>
<s>
Randolph	B-Application
diagrams	I-Application
can	O
be	O
interpreted	O
most	O
easily	O
by	O
defining	O
each	O
line	O
as	O
belonging	O
to	O
or	O
relating	O
to	O
one	O
logical	O
statement	O
or	O
set	O
.	O
</s>
<s>
Though	O
Venn	B-Application
diagrams	I-Application
are	O
more	O
commonly	O
used	O
to	O
represent	O
combinations	O
of	O
sets	O
,	O
Randolph	B-Application
diagrams	I-Application
have	O
the	O
advantage	O
of	O
being	O
able	O
to	O
cleanly	O
represent	O
combinations	O
of	O
more	O
than	O
3	O
sets	O
.	O
</s>
<s>
Venn	B-Application
diagrams	I-Application
require	O
either	O
extension	O
into	O
higher	O
spatial	O
dimensions	O
or	O
the	O
use	O
of	O
more	O
complicated	O
shapes	O
while	O
Randolph	B-Application
diagrams	I-Application
evenly	O
subdivide	O
for	O
every	O
additional	O
set	O
.	O
</s>
<s>
Here	O
is	O
a	O
comparison	O
between	O
a	O
Venn	B-Application
diagram	I-Application
and	O
R-diagram	B-Application
for	O
5	O
sets	O
of	O
logical	O
statements	O
:	O
</s>
<s>
Randolph	O
modified	O
McCulloch	O
's	O
system	O
with	O
a	O
new	O
way	O
of	O
representing	O
combinations	O
and	O
relationships	O
of	O
more	O
than	O
two	O
logical	O
statements	O
or	O
sets	O
,	O
namely	O
subdividing	O
each	O
section	O
of	O
the	O
R-diagram	B-Application
with	O
a	O
new	O
diagonal	O
line	O
for	O
each	O
new	O
element	O
introduced	O
.	O
</s>
<s>
Randolph	O
's	O
paper	O
suggests	O
that	O
his	O
original	O
notion	O
was	O
to	O
use	O
R-diagrams	B-Application
to	O
represent	O
logical	O
relationships	O
,	O
and	O
then	O
expanded	O
the	O
idea	O
to	O
be	O
applied	O
to	O
set	O
theory	O
as	O
well	O
.	O
</s>
<s>
Throughout	O
the	O
paper	O
,	O
R-diagrams	B-Application
are	O
used	O
in	O
conjunction	O
with	O
normal	O
logical	O
and	O
set	O
binary	O
operation	O
symbols	O
.	O
</s>
<s>
When	O
applying	O
R-diagrams	B-Application
to	O
logic	O
theory	O
,	O
logical	O
statements	O
p	O
,	O
q	O
,	O
and	O
r	O
can	O
each	O
become	O
a	O
line	O
or	O
multiple	O
lines	O
to	O
visually	O
display	O
the	O
validity	O
of	O
each	O
element	O
in	O
a	O
larger	O
statement	O
.	O
</s>
<s>
The	O
R-diagrams	B-Application
for	O
p	O
and	O
q	O
are	O
shown	O
below	O
,	O
respectively	O
:	O
</s>
<s>
The	O
R-diagram	B-Application
for	O
r	O
is	O
shown	O
below	O
:	O
</s>
<s>
R-diagrams	B-Application
are	O
primarily	O
used	O
to	O
represent	O
logical	O
expressions	O
.	O
</s>
<s>
Given	O
a	O
logical	O
proposition	O
,	O
R-diagrams	B-Application
are	O
able	O
to	O
display	O
the	O
outcome	O
of	O
every	O
possible	O
true/false	O
variation	O
of	O
each	O
element	O
,	O
creating	O
an	O
alternative	O
way	O
to	O
represent	O
a	O
truth	O
table	O
.	O
</s>
<s>
All	O
the	O
basic	O
logical	O
operations	O
,	O
or	O
connectives	O
,	O
can	O
be	O
expressed	O
using	O
an	O
R-diagrams	B-Application
as	O
a	O
more	O
easily	O
readable	O
alternative	O
to	O
a	O
truth	O
table	O
,	O
as	O
is	O
shown	O
in	O
the	O
table	O
below	O
:	O
</s>
<s>
R-diagrams	B-Application
can	O
be	O
used	O
to	O
easily	O
simplify	O
complicated	O
logical	O
expressions	O
,	O
using	O
a	O
step-by-step	O
process	O
.	O
</s>
<s>
Using	O
order	O
of	O
operations	O
,	O
logical	O
operators	O
are	O
applied	O
to	O
R-diagrams	B-Application
in	O
the	O
proper	O
sequence	O
.	O
</s>
<s>
Finally	O
,	O
the	O
result	O
is	O
an	O
R-diagram	B-Application
that	O
can	O
be	O
converted	O
back	O
into	O
a	O
simpler	O
logical	O
expression	O
.	O
</s>
<s>
It	O
can	O
be	O
simplified	O
using	O
R-diagrams	B-Application
as	O
follows	O
:	O
</s>
<s>
Similarly	O
,	O
R-diagrams	B-Application
can	O
be	O
used	O
to	O
prove	O
or	O
disprove	O
logical	O
arguments	O
.	O
</s>
<s>
which	O
can	O
then	O
be	O
simplified	O
using	O
R-diagrams	B-Application
:	O
</s>
<s>
The	O
result	O
is	O
an	O
R-diagram	B-Application
in	O
which	O
every	O
space	O
has	O
a	O
dot	O
.	O
</s>
<s>
An	O
R-diagram	B-Application
in	O
which	O
no	O
space	O
has	O
a	O
dot	O
is	O
a	O
contradiction	O
,	O
a	O
statement	O
that	O
is	O
never	O
true	O
.	O
</s>
<s>
R-diagrams	B-Application
are	O
also	O
used	O
in	O
set	O
theory	O
,	O
as	O
an	O
alternative	O
to	O
Venn	B-Application
diagrams	I-Application
.	O
</s>
<s>
As	O
in	O
logic	O
,	O
basic	O
set	O
operations	O
can	O
be	O
represented	O
visually	O
using	O
R-diagrams	B-Application
:	O
</s>
<s>
R-diagrams	B-Application
illustrate	O
the	O
equivalence	O
between	O
the	O
set	O
theoretical	O
and	O
logical	O
concepts	O
:	O
intersection	O
in	O
set	O
theory	O
is	O
equivalent	O
to	O
conjunction	O
in	O
logic	O
,	O
and	O
set	O
theory	O
's	O
union	O
is	O
equivalent	O
to	O
the	O
logical	O
disjunction	O
.	O
</s>
