ECS 120: Introduction to the Theory of Computation
Homework 1
Due Apr 8, by 1pm in the homework box in Kemper 2131
Problem 1.
Let
A, B, C
be three sets. Prove the following:
(a)
A
\
B
=
A
\
(
B
∩
A
).
(b)
B
⊆
A
if and only if
A
∩
B
=
∅
.
(c) (
A
\
B
)
\
C
=(
A
\
C
)
\
(
B
\
C
)=
A
\
(
B
∪
C
).
(d)
A
∩
B
=
∅
and
A
∩
B
=
∅
if and only if
A
=
B
.
(a) Let
x
be any element from the set
A
\
B
.Then
x
∈
A
and
x
6∈
B
.Thu
s
x
6∈
A
∩
B
,
since it is not in both. But then
x
∈
A
\
(
B
∩
A
). So, all elements of
A
\
B
are
elements of
A
\
(
B
∩
A
)too
. Conv
e
r
se
ly
,le
t
x
∈
A
\
(
B
∩
A
). Then,
x
∈
A
and
x
6∈
A
∩
B
s
x
is in
A
but not
B
.S
o
,
x
∈
A
\
B
and therefore all elements of
A
\
(
B
∩
A
) are elements of
A
\
B
too.
(b) (If) If
A
∩
B
=
∅
and
x
∈
B
,then
x
∈
A
, i.e.
B
⊆
A
.
(Only if) If
B
⊆
A
and
x
∈
B
then
x
∈
A
. Then,
x
6∈
A
, and thus
A
∩
B
=
∅
.
(c) First we show that
A
\
B
=
A
∩
B
.Name
,i
f
x
∈
A
and
x
6∈
B
then
x
∈
A
and
x
∈
B
.T
h
u
s
x
∈
A
∩
B
.C
o
n
v
e
r
s
e
l
y
f
x
∈
A
∩
B
then
x
∈
A
and
x
∈
B
h
u
s
x
∈
A
and
x
6∈
B
, and the result follows.
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 Spring '07
 Filkov
 Elementary algebra, DFAS, ak k

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