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ECE 3250
HOMEWORK ASSIGNMENT I
Fall 2009
1.
Find, read, and understand a proof of the Schr¨
oderBernstein Theorem, which states:
Let
A
and
B
be sets. If there exists an injective mapping
f
:
A
→
B
and an injective
mapping
g
:
B
→
A
, then there exists a bijective mapping
h
:
A
→
B
. If you look online,
it might help to search for “SchroederBernstein.”
2.
Given a set
A
, ﬁrst let
P
o
(
A
) be the power set of
A
without the empty set — i.e.,
P
o
(
A
) is the set of all nonempty subsets of
A
. A mapping
κ
:
P
o
(
A
)
→
A
is called a
choice function
if
κ
(
S
)
∈
S
for all
S
∈ P
o
(
A
). In other words, a choice function “picks
out” an element of every nonempty subset of
A
. Show that a choice function cannot be
an injective mapping if
A
has at least two elements. (Please don’t do this by referring to
the relative cardinalities of the sets — it’s easier than that. What’s more interesting is
the question: does a choice function exist for every set
A
? The
Axiom of Choice
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