Math 240, Fall 2009 — Assignment 2 — Due: Monday, October 5
1. Suppose that five ones and four zeros are arranged around a circle. Let’s use the word
bit
to mean either a zero or a one. We do the following steps:
Step 1: Between any two equal bits, insert a 0 and between any two unequal bits, insert
a 1. This produces nine new bits.
Step 2: Erase the nine original bits.
Prove that when we perform this procedure repeatedly, we never have nine zeros around
the circle after Step 2. Hint: Give a proof by contradiction. Work backward, assuming
that you did end up with nine zeros.
2. Suppose that
A
,
B
,
C
are sets. Prove that (
B

A
)
∪
(
C

A
) = (
B
∪
C
)

A
.
3. Suppose
A
and
B
are two finite sets. Recall that
P
(
A
) is the
power set
of
A
, which is
the set of all subsets of
A
.
(a) Prove, or disprove, that
P
(
A
∩
B
) =
P
(
A
)
∩
P
(
B
).
(b) Prove, or disprove, that
P
(
A
∪
B
) =
P
(
A
)
∪
P
(
B
).
4. Each of the following functions
f
is a function from
R
to
R
. Determine whether each
f
is injective (onetoone), surjective (onto), or both. Please give reasons.
(a)
f
(
x
) =

3
x
+ 4
(b)
f
(
x
) =

3
x
2
+ 7
(c)
f
(
x
) =
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 Spring '08
 SZABO
 Math, Inverse function, smallest possible order, Jonathan Cottrell Instructor, Benjamin Young Date

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