MATH 74 HOMEWORK 2
1
Due Wednesday February 4th at 3:10pm.
(1)
(a) Define the statement ‘
f
:
R
→
R
is continuous at
a
∈
R
.’
(b) Write the negation of the statement in (a), without using ‘not.’
(c) Choose an example of a function that is not continuous at a point and prove that it is discon
tinuous using the definition from (b).
(2) (Bergman 6) Suppose that
A
and
B
are sets.
Translate each of the statements below, which are
expressed using the symbols
∀
and
∃
into a statement about
A
and
B
expressed using only the
settheoretic symbols discussed in class (
⊂
,
,
,
∅
,
,
c
, etc.).
(a) (
∀
x
)((
x
∈
A
) =
⇒
(
x
∈
B
))
(b) (
∀
x
)((
x
∈
A
)
⇐⇒
(
x
∈
B
))
(c) (
∀
x
)(
x /
∈
A
)
(d) (
∃
x
)((
x /
∈
A
)
∧
(
x
∈
B
)).
(3) (Bergman 9) Consider the sentence, “There is someone at the hotel who cleans every room.” Explain
two ways this sentence can be interpreted and translate them into two quantifications of the relation
“X cleans Y.”
(4) Let
S
= [0
,
2] and let
T
=
{
6
/n

n
∈
Z
and
n
≥
1
}
.
Compute, with proof, (
S
×
T
)
(
Z
×
Z
).
(5) Prove the 2nd De Morgan Law: (
A
B
)
c
=
A
c
B
c
.
(6) Determine (with proof) if the following functions are injective, surjective, both or neither.
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 Spring '09
 DANBERWICK
 Math, Division, Cartesian product, Bergman, Morgan Law, settheoretic symbols, statement ‘f

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