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Nicola Hodges
David Walnut
MATH 290002
September 28, 2010
Writing Assignment 3
1.
Prove that any sets A and B, (A – B) U (B – A) = (A U B) – (A
∩
B)
First we will show (A – B) U (B – A)
⊆
(A U B) – (A
∩
B).
Assume x
∈
[(A – B) U (B – A) ].
So x
∈
A
and
X
∉
B or x
∈
B
and X
∉
A.
First assume x
∈
A
and hence X
∉
B.
So x
∈
(A U B) and x
∉
(A
∩
B)
So, x
∈
(A U B) – (A
∩
B).
Second, assume x
∈
B and hence X
∉
A.
So x
∈
(A U B) and x
∉
(A
∩
B)
So, x
∈
(A U B) – (A
∩
B).
Therefore, (A – B) U (B – A)
⊆
(A U B) – (A
∩
B).
Next we will show that (A U B) – (A
∩
B)
⊆
(A – B) U (B – A).
Assume x
∈
(A U B) and x
∉
(A
∩
B)
So x
∈
A or x
∈
B, but x
∉
A and B.
First assume x
∈
A, and
Since x
∉
(A
∩
B), x
∉
B.
Hence, x
∈
(A – B) and x
∉
(B – A)
So, x
∈
(A – B) U (B – A)
Second, assume x
∈
B
Since, x
∉
(A
∩
B), x
∉
A.
Hence, x
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 Advanced Math, Sets

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