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39
2.2
If
A
and
B
are subsets of a set
X
, prove that
A
−
B
=
A
∩
B
0
, where
B
0
=
X
−
B
is the complement of
B
.
Solution.
This is one of the beginning set theory exercises that is so easy
it is dif
f
cult; the dif
f
culty is that the whole proof turns on the meaning of
the words
“
and
”
and
“
not.
”
For example, let us prove that
A
−
B
⊆
A
∩
B
0
.
If
x
∈
A
−
B
, then
x
∈
A
and
x
/
∈
B
; hence,
x
∈
A
and
x
∈
B
0
, and so
x
∈
A
∩
B
0
. The proof is completed by proving the reverse inclusion.
2.3
Let
A
and
B
be subsets of a set
X
. Prove the
de Morgan laws
(
A
∪
B
)
0
=
A
0
∩
B
0
and
(
A
∩
B
)
0
=
A
0
∪
B
0
,
where
A
0
=
X
−
A
denotes the complement of
A
.
Solution.
Absent.
2.4
If
A
and
B
are subsets of a set
X
,de
f
ne their
symmetric difference
(see
Figure 2.5) by
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.
 Fall '11
 KeithCornell

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