This preview shows pages 1–2. Sign up to view the full content.
Homework 1, Math 245, Fall 2010
Due Wednesday, September 15 in class.
The solution of each exercise should be at most one page long. If you can, try to write
your solutions in LaTex. Each question is worth 2 points.
1. Let
A,B,C
be sets. Prove that
A
∩
(
B
\
C
) = (
A
∩
B
)
\
(
A
∩
C
).
Proof.
We prove this statement by double inclusion.
First, we show that
A
∩
(
B
\
C
)
⊂
(
A
∩
B
)
\
(
A
∩
C
).
Let
x
∈
A
∩
(
B
\
C
). This means
x
∈
A
and
x
∈
B
\
C
. The last statement implies
x
∈
B
and
x /
∈
C
. Now, as
x
∈
A
and
x
∈
B
, it follows that
x
∈
A
∩
B
. Also, as
x
∈
A
and
x /
∈
C
, we deduce that
x /
∈
A
∩
C
(otherwise,
x
∈
A
∩
C
would imply
x
∈
C
which is
a contradiction with
x /
∈
C
). Thus, we obtained that
x
∈
A
∩
B
and
x /
∈
A
∩
C
. Hence,
x
∈
(
A
∩
B
)
\
(
A
∩
C
) which shows
A
∩
(
B
\
C
)
⊂
(
A
∩
B
)
\
(
A
∩
C
).
Now we prove the opposite inclusion (
A
∩
B
)
\
(
A
∩
C
)
⊂
A
∩
(
B
\
C
).
Let
y
∈
(
A
∩
B
)
\
(
A
∩
C
). This means
y
∈
A
∩
B
and
y /
∈
A
∩
C
. The fact that
x
∈
A
∩
B
means that
x
∈
A
and
x
∈
B
. Now as
y
∈
A
and
y /
∈
A
∩
C
, we deduce that
y /
∈
C
(otherwise,
y
∈
C
combined with
y
∈
A
would imply
y
∈
A
∩
C
which is a contradiction
with
y /
∈
A
∩
C
). Thus, we have proved that
y
∈
A,y
∈
B
and
y /
∈
C
. The last two
facts imply
y
∈
B
\
C
. Combining this with
y
∈
A
shows that
y
∈
A
∩
(
B
\
C
). Hence,
we have proved that (
A
∩
B
)
\
(
A
∩
C
)
⊂
A
∩
(
B
\
C
). This ﬁnishes the proof of our
exercise.
2. Let
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '10
 CIOABA
 Sets

Click to edit the document details