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Math 5020 Homework 5
due Friday, April 8, 2011
Assume ZFC
−
Regularity.
1. Let
X
and
Y
be sets. Show that
(a)

X

=

Y

iﬀ

X

e
=

Y

e
.
(b)

X
 ≤ 
Y

iﬀ

X

e
≤ 
Y

e
.
(c)

X
 ≤ 
Y

iﬀ there is a function
f
:
Y
→
X
with rng(
f
) =
X
.
2. Show that the set of all functions
f
:
R
→
R
has cardinality 2
2
ℵ
0
. (
Hint
: You may use the fact

R

=
P
(
ω
)

.)
3. Show that the set of all continuous functions
f
:
R
→
R
has cardinality 2
ℵ
0
. (
Hint
: A continuous
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 Spring '11
 staff
 Set Theory, Sets

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