CS153 Autumn 2009: Homework 1
Due Tuesday, October 13th at 3PM
1.
(10 points)
Prove that proof by contrapositive is valid.
In other words, show that the
statements “if
A
then
B
” (
A
→
B) and “if not
B
then not
A
” (
¬
B
→ ¬
A
) are equivalent by
using a truth table.
2.
(6 points)
Prove the following statement about integers: if
a
2
is even, then
a
is even.
3.
(10 points) Based on Hein, Page 34, Exercise 24
In lecture, we started the proof that “power(
A
∩
B
) = power(
A
)
∩
power(
B
)” by showing that
“power(
A
∩
B
)
⊂
power(
A
)
∩
power(
B
)”. Complete the proof by showing that “power(
A
)
∩
power(
B
)
⊂
power(
A
∩
B
)”.
4.
(6 points)
Write down the converse of the following statement about integers: “If
x
and
y
sum to 10, then they must have the same parity.”
5.
(8 points) Hein page 32, Exercise 10, Parts b, d, f, and h.
For each integer
i
define
A
i
as follows:
•
If
i
is even then
A
i
=
{
x

x
∈
Z
and
x <

i
or
i < x
}
.
•
If
i
is odd then
A
i
=
{
x

x
∈
Z
and

i < x < i
}
.
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
 staff
 ice cream, If and only if, Garden strawberry, Hein

Click to edit the document details