CSCE 235
Homework
Assignment 1
Assigned: February 14, 2008
Due: 12:30 p.m. February 23,
2008
(Homework 5
minutes late will not be accepted)
Logic Basics
1. (3 points) Consider the following propositions:
, 
, 
, 
, 
, 
, 
, 
(a) Which proposition is the converse of ?
(b) Which
proposition is the contrapositive of
?
(c) Which
propositions are logically equivalent to ?
2. (3 points) Suppose that is known to be
false. Give the truth values for
(a) 
(b) 
(c)
3. (5 points) Construct the truth table for
(a) 
(b) 
(c) 
(d) 
(e) 
Logic Equivalences and Implications
4. (5 points) Verify the following logical implications using truth tables.
(a) 
, modus tollens
(b) 
, disjunctive syllogism
5. (10 points) Prove or disprove the following. (Hint: only one line of the truth table is needed to show that a proposition is not a tautology.)
(a) 
(b) 
(c) 
(d) 
(e) 
6. (3
points) The
“exclusive or” connective 
is defined by the
truth table:
|
p |
q |
|
|
0 |
0 |
0 |
|
0 |
1 |
1 |
|
1 |
0 |
1 |
|
1 |
1 |
0 |
(a) Show
that 
.
(b) Show that 
.
7. (7 points) Every compound
proposition can be written using only the connectives 
and 
. This fact follows
from the equivalences 
, 
, and 
. Find propositions
logically equivalent to the following using only the connectives and .
(a) 
(b) 
(c) 
(d) 
(Hint: see problem
6(b).)
8. (4 points) The Sheffer Stroke is a connective 
defined by the truth table:
|
p |
q |
|
|
0 |
0 |
1 |
|
0 |
1 |
1 |
|
1 |
0 |
1 |
|
1 |
1 |
0 |
This connective is interesting because all compound propositions can be written using only this connective. You may prove logically or use a truth table (or a combination of both) for the following.
(a) Show
that 
.
(b) Show that 
.
(c) Find
a proposition equivalent to using only the Sheffer Stroke.
Formal Proofs
9. (10 points) Complete the following formal proofs by supplying explanations for each step.
(a) If 
, 
and 
, then 
.
Proof
1.
2.
3.
4. 
5.
(b) If 
, 
and 
, then 
.
Proof
1.
2.
3.
4. 
5.
6. 
7. 
8.
(c) If 
, 
and 
, then 
.
Proof
1.
2.
3.
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11.
10. (10 points) Complete the following proof by contradiction by supplying explanations for each step.
(a) If , and , then .
Proof
1.
2.
3.
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11. 
12. 
13.
14. 
15. contradiction
(b) If , 
and 
, then 
.
Proof
1.
2.
3.
4. 
5.
6.
7.
8. 
9. 
10. 
11. 
12. 
13. contradiction
•
Based on Ross and Wright (1988).