<s>
Double	O
complement	O
or	O
involution	B-Algorithm
law	O
:	O
</s>
<s>
Set	O
subtraction	O
is	O
quasi-commutative	B-Algorithm
and	O
satisfies	O
the	O
Jordan	O
identity	O
.	O
</s>
<s>
For	O
symmetric	O
difference	O
,	O
the	O
sets	O
and	O
are	O
always	O
disjoint	B-Algorithm
.	O
</s>
<s>
Quasi-commutativity	B-Algorithm
:	O
</s>
<s>
Any	O
given	O
indexed	O
family	O
of	O
sets	O
(	O
which	O
is	O
a	O
function	O
)	O
can	O
be	O
canonically	O
associated	O
with	O
its	O
image/range	O
(	O
which	O
is	O
a	O
family	O
of	O
sets	O
)	O
.	O
</s>
<s>
However	O
,	O
this	O
is	O
a	O
bijective	B-Algorithm
correspondence	O
because	O
an	O
indexed	O
family	O
of	O
sets	O
is	O
required	O
to	O
be	O
injective	O
(	O
that	O
is	O
,	O
there	O
may	O
exist	O
distinct	O
indices	O
such	O
as	O
)	O
,	O
which	O
in	O
particular	O
means	O
that	O
it	O
is	O
possible	O
for	O
distinct	O
indexed	O
families	O
of	O
sets	O
(	O
which	O
are	O
functions	O
)	O
to	O
be	O
associated	O
with	O
the	O
same	O
family	O
of	O
sets	O
(	O
by	O
having	O
the	O
same	O
image/range	O
)	O
.	O
</s>
<s>
If	O
all	O
are	O
pairwise	O
disjoint	B-Algorithm
and	O
all	O
are	O
also	O
pairwise	O
disjoint	B-Algorithm
,	O
then	O
so	O
are	O
all	O
(	O
that	O
is	O
,	O
if	O
then	O
)	O
.	O
</s>
<s>
:	O
In	O
the	O
particular	O
case	O
where	O
all	O
are	O
equal	O
(	O
that	O
is	O
,	O
for	O
all	O
which	O
is	O
the	O
case	O
with	O
the	O
family	O
for	O
example	O
)	O
,	O
then	O
letting	O
denote	O
this	O
common	O
set	O
,	O
the	O
Cartesian	O
product	O
will	O
be	O
which	O
is	O
the	O
set	B-Algorithm
of	I-Algorithm
all	I-Algorithm
functions	I-Algorithm
of	O
the	O
form	O
The	O
above	O
set	O
equalities	O
and	O
,	O
respectively	O
become	O
:	O
</s>
<s>
on	O
the	O
right	O
hand	O
side	O
,	O
the	O
indices	O
range	B-Algorithm
over	O
(	O
so	O
the	O
subscripts	O
of	O
range	B-Algorithm
over	O
)	O
.	O
</s>
<s>
Every	O
map	O
can	O
be	O
bijectively	B-Algorithm
identified	O
with	O
the	O
pair	O
(	O
the	O
inverse	O
sends	O
to	O
the	O
map	O
defined	O
by	O
and	O
this	O
is	O
technically	O
just	O
a	O
change	O
of	O
notation	O
)	O
.	O
</s>
<s>
However	O
,	O
sometimes	O
these	O
products	O
are	O
somehow	O
identified	O
as	O
the	O
same	O
set	O
through	O
some	O
bijection	B-Algorithm
or	O
one	O
of	O
these	O
products	O
is	O
identified	O
as	O
a	O
subset	O
of	O
the	O
other	O
via	O
some	O
injective	O
map	O
,	O
in	O
which	O
case	O
(	O
by	O
abuse	O
of	O
notation	O
)	O
this	O
intersection	O
may	O
be	O
equal	O
to	O
some	O
other	O
(	O
possibly	O
non-empty	O
)	O
set	O
.	O
</s>
<s>
For	O
another	O
example	O
,	O
take	O
and	O
with	O
and	O
all	O
equal	O
to	O
Then	O
and	O
which	O
can	O
both	O
be	O
identified	O
as	O
the	O
same	O
set	O
via	O
the	O
bijection	B-Algorithm
that	O
sends	O
to	O
Under	O
this	O
identification	O
,	O
</s>
<s>
Mnemonic	O
:	O
In	O
fact	O
,	O
for	O
each	O
of	O
the	O
above	O
four	O
set	O
formulas	O
for	O
which	O
equality	O
is	O
not	O
guaranteed	O
,	O
the	O
direction	O
of	O
the	O
containment	O
(	O
that	O
is	O
,	O
whether	O
to	O
use	O
)	O
can	O
always	O
be	O
deduced	O
by	O
imagining	O
the	O
function	O
as	O
being	O
and	O
the	O
two	O
sets	O
(	O
and	O
)	O
as	O
being	O
non-empty	O
disjoint	B-Algorithm
subsets	O
of	O
its	O
domain	O
.	O
</s>
<s>
The	O
answer	O
to	O
such	O
a	O
question	O
can	O
,	O
as	O
before	O
,	O
be	O
deduced	O
by	O
consideration	O
of	O
this	O
constant	O
function	O
:	O
the	O
answer	O
for	O
the	O
general	O
case	O
(	O
that	O
is	O
,	O
for	O
arbitrary	O
and	O
)	O
is	O
always	O
the	O
same	O
as	O
the	O
answer	O
for	O
this	O
choice	O
of	O
(	O
constant	O
)	O
function	O
and	O
disjoint	B-Algorithm
non-empty	O
sets	O
.	O
</s>
<s>
For	O
statement	O
(	O
d	O
)	O
,	O
this	O
is	O
the	O
same	O
as	O
:	O
"	O
for	O
all	O
singleton	O
subsets	O
"	O
(	O
because	O
the	O
definition	O
of	O
"	O
pairwise	O
disjoint	B-Algorithm
"	O
is	O
satisfies	O
vacuously	O
by	O
any	O
family	O
that	O
consists	O
of	O
exactly	O
1	O
set	O
)	O
.	O
</s>
<s>
Then	O
and	O
is	O
a	O
sequence	O
of	O
pairwise	O
disjoint	B-Algorithm
sets	I-Algorithm
.	O
</s>
<s>
Suppose	O
that	O
is	O
non-decreasing	O
,	O
let	O
and	O
let	O
for	O
every	O
Then	O
and	O
is	O
a	O
sequence	O
of	O
pairwise	O
disjoint	B-Algorithm
sets	I-Algorithm
.	O
</s>
