CSCE 235
Homework
Assignment 1
Assigned: January 12, 2009
Due: 12:30 p.m. January 21,
2009
(Homework 5 minutes late will not be
accepted)
Logic Basics
1. (5 points) Consider the following propositions:
, 
, 
, 
, 
, 
, 
, 
,
(a) Which
proposition is the converse of 
? (Identify all equivalences)
(b) Which
proposition is the contrapositive of
? (Identify all equivalences)
2. (5 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) 
, addition
(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. (5
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. (10 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. (10 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. (15 points) Complete the following formal proofs by supplying explanations for each step.
(a) If 
, 
and 
, then 
.
Proof Explanations
1.
2.
3.
4. 
5. 
6. 
7.
(b) If 
, 
, 
, then 
.
Proof Explanations
1.
2.
3.
4. 
5. 
6. 
7. 
8. 
9. 
10.
(c) If 
, 
and 
, then 
.
Proof Explanations
1.
2.
3.
4. 
5. 
6.
7.
8. 
10. (20 points) Complete the following proof by contradiction by supplying explanations for each step.
(a) If , , , then .
Proof Explanations
1.
2.
3.
4. 
5. 
6. 
7. 
8. 
9. 
10. 
11.
12.
13.
14.
15. 
16. contradiction
(b) If 
, 
, , and 
, then .
Proof Explanations
1.
2.
3.
4.
5. 
6. 
7. 
8. 
9. 
10.
14. 
15. 
16. contradiction
•
Based on Ross and Wright (1988).