1. Give an example of an axiom from Euclidean geometry.
2. Give an example of a theroem from Euclidean geometry. What is the
difference between an axiom and a theorem?
3. Assume we already know that if a, b, and c are real numbers with
ab = ac and a ¹ 0, then b = c. Justify
each step in the following proof that if xy = 0, then either x = 0 or y
= 0.
Pf.: Suppose that xy = 0 and x ¹
0 and y ¹ 0.
Since xy = 0 = x * 0
And x ¹
0,
then y = 0,
which is a contradiction.
Therefore either x or y
must equal 0.
Write each of 4 through 7 symbolically and determine their validity.
Use the propositions
p: I study hard.
q. I get A's
r: I get rich.
4. If I study hard, then I get A's.
I study hard.
\ I get A's.
5. If I study hard or I get rich, then I get A's.
I get A's.
\ If I don't study hard,
then I get rich.
6. I study hard if and only if I get rich.
I get rich.
\ I study hard.
7. If I study hard, then I get A's or I get rich.
I don't get A's and I don't get rich.
\ I don't study hard.
8. What is wrong with the following argument?
Nothing is better than perfect happiness.
A ham sandwich is better than nothing.
\ A ham sandwich is better
than perfect happiness.
For each of the following, let
p = My computer is slow.
q = I will upgade my computer.
r = I will buy a new computer.
Write each of the following in words and determine the validity of
the argument.
9. p ® r
p ® q
\ p ®
(r Ù q)
10. p ® r
r ®
q
\
q
11. Ø r ®Ø
p
r
\ p
12. p ®
q
Ø
p
\ Ø
q
13. p «
q
Ø
p
\ Ø
q
14. p ® (q ®
r)
q ®
(p ® r)
\
(p Ú q) ®
r