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Sizes of sets
Paul Kantor
1 Size of the Union of Two sets
We have shown that the size of the union of two sets can be expressed in terms of the
sizes of the sets, and the size of the
intersection
of the two sets.
1.1 Rigorous mathematical argument
We did this by starting from the obvious fact (we could say that this is an axiom about the
sizes of sets) that if two sets have no overlap, then we can write

 





"
"
A
B
A
B
where S means thesizeof the set S
Then we noted that for any set A, and any other set B, we can write that:
(
)
(
')
'
A
A
B
A
B
where B is the complement of B
Now we also know that this way of writing the set A represents it as the union of two sets
that have no overlap, because
(
)
(
(
)
(
: (
(
)
(
A
B
A
B
A
A
B
B
and the underlined part
B
B
so
A
B
A
B
So, armed with this, we know that

  (
)
 (
')
A
A
B
A
B
And, by exactly the same reasoning, we can show that:

  (
 ( '
B
A
B
A
B
So we are about halfway through the analysis at this point.
The next thing we do is
combine the breakdown of A and the breakdown of B:
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)
(
')
(
)
( '
)
[(
)
(
')]
[(
)
( '
)]
(
)
(
)
(
)
(
)
(
A
A
B
A
B
and B
A
B
A
B
so
A
B
A
B
A
B
A
B
A
B
then we rearrange these four pieces and note that
A
B
A
B
A
B
so A
B
A
B
A
B
( '
)]
AB
which is the union of
non overlapping sets
3
But for nonoverlapping pieces, we know how to figure out the size. So we find

  (
)
 (
')
 ( '
A
B
A
B
A
B
A
B
We are almost done. Let’s add the formulas for the sizes of the two sets separately:

  (
 (
A
A
B
A
B

  (
) 
 (
') 

  (
) 
 ( '
) 




 (
) 
 (
') 
 ( '
) 
A
A
B
A
B
B
A
B
A
B
A
B
A
B
A
B
A
B
2
This is
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 Fall '09
 Boros

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