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CS 341 Automata Theory
Elaine Rich
Homework 1
Due Thursday, September 7 at 11:00 a. m.
1)
Write each of the following explicitly:
a)
P
({
a
,
b
}) –
P
({
a
,
c
})
b)
{
a
,
b
}
×
{1, 2, 3}
×
∅
c)
{
x
∈
ℕ
: (
x
≤
7
∧
x
≥
7}
d)
{
x
∈
ℕ
:
5
y
∈
ℕ
(
y
< 10
∧
(
y
+ 2 =
x
))} (where
ℕ
is the set of nonnegative integers)
e)
{
x
∈
ℕ
:
5
y
∈
ℕ
(
5
z
∈
ℕ
((
x
=
y
+
z
)
∧
(
y
< 5)
∧
(
z
< 4)))}
2)
Give an example, other than one of the ones in the book, of a relation on the set of people that is reflexive and
symmetric but not transitive.
3)
Define
≡
p
to be “equivalent modulo
p
”.
x
≡
p
y
iff
x
modulo
p
=
y
mod
p
.
For example 4
≡
3
7 and
7
≡
5
12.
Let
R
p
be a binary relation on
ℕ
, defined as follows, for any
p
≥
1:
R
p
= {(
a
,
b
):
a
≡
p
b
}
So, for example
R
3
contains (0, 0), (6, 9), (1, 4), etc., but does not contain (0, 1), (3, 4), etc.
a)
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 Fall '08
 Rich

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