MATH 3283W. Sequences, Series, and Foundations:
Writing Intensive. Spring 2009
Homework 1. Problems and Solutions
I. Writing Intensive Part
1
(5 points). Check whether or not each of the following statements can
be true for some values (”true” or ”false”) of
P
and
Q
. Write out the truth
table for each of these statements.
A
= (
P
=
⇒
Q
)&(
P
=
⇒ ¬
Q
)
,
B
= (
P
=
⇒
Q
)&(
¬
P
=
⇒
Q
)
,
C
= (
P
=
⇒
Q
)&(
¬
P
=
⇒ ¬
Q
)
.
Solution.
One can compose the truth tables for
A, B, C
by direct con
sideration of four possible cases: (
P, Q
) = (
T, T
)
,
(
T, F
)
,
(
F, T
)
,
and (
F, F
)
.
Here the equality (
P
=
⇒
Q
) = (
¬
P
∨
Q
) may be helpful. Some simplifica
tions are possible:
(i)
If
P
=
T,
then one of implications
P
=
⇒
Q
or
P
=
⇒ ¬
Q
is false,
because either
Q
or
¬
Q
is false. Therefore, in this case
A
=
F
. If
P
=
F
,
then both implications are true (footnote 7 on p.2 in the textbook), so that
A
=
T.
This means that we always have
A
=
¬
P
.
(ii)
If
Q
=
T,
then both implications are true, and
B
=
T
. If
Q
=
F,
then one of implications is false, and
B
=
F.
In any case,
B
=
Q.
(iii)
Since (
¬
P
=
⇒ ¬
Q
) is equivalent to (
Q
=
⇒
P
) (p.5 in the textbook),
we have
C
= (
P
⇐⇒
Q
)
.
2
(5 points). Prove that a subset of
N
is bounded if and only if it is finite.
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 Spring '09
 Math, Natural number, n1 n n1

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