EECS 20. Midterm No. 1
October 4, 2002.
Solution
1.
50 points.
Please indicate whether the following statements are true or false. There will be
no partial credit. They are either true or false. So please be sure of your answer.
(a)
∀
t
∈
Reals
,
(
t, t
+1)
∈
Reals
2
true
(b)
∃
x
∈
Integers
,
{
(
x, x
+1)
}⊂{
1
,
2
,
3
}
2
true
(c) If
A
=
{
1
,
2
}
and
B
=
{
1
,
2
,
3
}
, then
∃
x
∈
A
such that
∀
y
∈
B
,
x<
=
y
.
true
(d)
P
(
A
∪
B
)=
P
(
A
)
∪
P
(
B
)
, where
P
denotes the power set.
false
(e) For any two functions
f
:
A
→
A
and
g
:
A
→
A
, where
A
is a set,
f
◦
g
=
g
◦
f
.
false
(f) Let
f
:
Reals
→
Reals
be a function where
∀
x
∈
Reals
,
f
(
x
)=
x
sin(
x
)
. Then
f
is
onto.
true
(g) For the same function
f
in the previous part,
f
is one-to-one.
false
(h) Let
A
=[
−
1
,
1]
. Consider a function
f
where
∀
x
∈
A
,
f
(
x
)=
x
sin(2
πx
)
. Then
f
∈
[
A
→
A
]
.
true
(i)
[
{
1
,
2
,
3
}→{
1
,
2
}
]
⊂
[
{
1
,
2
,
3
}→
Naturals