CSE 20: Discrete Mathematics
Fall 2011
Problem Set 5
Instructor: Daniele Micciancio
Due on:
Fri. Nov. 11, 2011
Problem 1 (8 points)
Remember, the image of a set
X
⊆
A
under a function
f
:
A
→
B
is the set
f
(
X
) =
{
f
(
x
)

x
∈
X
}
, while
the inverse image of a set
Y
⊆
B
is the set
f

1
(
Y
) =
{
x
∈
A

f
(
x
)
∈
Y
}
.
In class we asked if for every set
X
⊆
A
and function
f
:
A
→
B
, it is true that
f

1
(
f
(
X
)) =
X
, and we
began answering this question by proving that
X
⊆
f

1
(
f
(
X
)). We left open question whether the reverse
inclusion
X
⊇
f

1
(
f
(
X
)) also holds.
Prove or disprove that for every
f
:
A
→
B
and
X
⊆
A
, it holds that
X
⊇
f

1
(
f
(
X
)) also holds. Your
answer should contain:
1. A clear claim of which statement you are proving. (This can be the given statement or its negation.
2. A proof that your claim is correct.
Problem 2 (16 points)
In this problem you are asked to prove that for any function
f
:
A
→
B
and set
Y
⊆
B
, if
f
is onto, then
f
(
f

1
(
Y
)) =
Y
. Structure your proof as follows:
•
Prove as a first lemma that
f
(
f

1
(
Y
))
⊇
Y
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Foster
 Daniele Micciancio, ﬁrst lemma

Click to edit the document details