MATH 74 HOMEWORK 6 (DUE WEDNESDAY OCTOBER 17)
Assume a calculuslevel familiarity with arithmetic in this problem set.
1.
Let
A
=
{
a
2

b
2
:
a,b
∈
Z
}
and
B
=
{
2
k

1 :
k
∈
Z
}
.
1(a).
Is
A
⊆
B
?
If yes, prove it.
If no, explain why.
(This means: give a
counterexample, and as much discussion that a casual reader might need to be
convinced that it is a counterexample.)
1(b).
Is
B
⊆
A
? If yes, prove it. If no, explain why.
1(c).
Is
A
=
B
? If yes, explain why. If no, explain why.
2.
Let
A
,
B
, and
C
be sets.
2(a).
Suppose that
A
⊆
B
. Is it necessarily also true that then
A
∩
B
⊆
B
∩
C
? If
so, prove it. If not, explain why.
2(b).
Suppose that
A
∩
C
⊆
B
∩
C
. Is it necessarily also true that
A
⊆
B
? If so,
prove it. If not, explain why.
3.
Let
A
,
B
, and
C
be sets. Suppose that
A
∩
B
and
B
∩
C
are nonempty. Is it
necessarily also true that
A
∩
C
is nonempty? If so, prove it. If not, explain why.
4.
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 Fall '07
 COURTNEY
 Calculus, Empty set

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