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Proving two sets are equal
In general to show that
X
=
Y
we need to show two things:
X
⊆
Y
and
Y
⊆
X
Occasionally, however, the proof follows directly by logical equivalence:
Example
(DeMorgan’s Law):
A
∪
B
=
A
∩
B
Proof
:
A
∪
B
=
{
x
∈
U
x
∉
A
∪
B
}
=
{
x
∈
U
¬
(
x
∈
A
∪
B
) }
=
{
x
∈
U
¬
(
x
∈
A
∨
x
∈
B
) }
=
{
x
∈
U
¬
(
x
∈
A
)
∧ ¬
(
x
∈
B
) }
=
{
x
∈
U
(
x
∉
A
)
∧
(
x
∉
B
) }
=
{
x
∈
U
(
x
∈
A
)
∧
(
x
∈
B
) }
=
A
∩
B
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Claim
:
A
=
(
A
−
B
)
∪
(
A
∩
B
)
Proof
:
Let
x
∈
A
There are two cases:
x
∈
B
and
x
∉
B
If
x
∈
B
,
then
(
x
∈
A
)
∧
(
x
∈
B
)
and hence
x
∈
A
∩
B
Thus
x
∈
(
A
−
B
)
∪
(
A
∩
B
)
If
x
∉
B
, then
(
x
∈
A
)
∧
(
x
∉
B
)
and hence
x
∈
(
A
−
B
)
Thus
x
∈
(
A
−
B
)
∪
(
A
∩
B
)
We have shown that
A
⊆
(
A
−
B
)
∪
(
A
∩
B
)
Now suppose
x
∈
(
A
−
B
)
∪
(
A
∩
B
),
so
(
x
∈
A
−
B
)
or
(
x
∈
A
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