1. August 31, 2005

2. Homework Problem Some information about homework problem.
C'  is the conclusion - it is NOT a hypothesis The problem CAN be done using our original statements (i.e. without replacing T by K) Here’s what the lawyer said
If my client is guilty, then the knife was in the drawer
Either the knife was not in the drawer or Jason Pritchard saw the knife.
If the knife was not there on October 10, it follows that Jason Pritchard did not see the knife.
Furthermore, if the knife was there on October 10, then the knife was in the drawer and also the hammer was in the barn.
But we all know that the hammer was not in the barn.
Therefore, ladies and gentlemen of the jury, my client is innocent.

3. Here are the variables used
C client is guilty K knife was in the drawer T knife was there on October 10 S Jason Pritchard saw knife H hammer was in the barn
1. C -> K hypothesis 2. K' v  S hypothesis 3. T' -> S' hypothesis 4. T -> (K ^ H) hypothesis 5. H' hypothesis

4. Proof 6. H’ v K’ 5. addition 7. (H ^ K)’ 6. DeMorgan’s
8. (K ^ H)’ 7. commutative
9. T’ 4,8 Modus Tollens
10. S’ 3,9 Modus Ponens
11. S v K’ 2 commutative
12 K’ 10,11 disjunctive syllogism
13 C’ 1,12 Modus Tollens

5. Statement of problem If the program is efficient, it executes quickly
Either the program is efficient, or it has a bug.
However, the program does not execute quickly.
Therefore it has a bug.
Please use the letters E,Q,B

6. Propositional form
(E → Q) ⋀ (E ⋁ B) ⋀ Q’→ B

7. A solution 1. E → Q hyp (simplification) 2. E ⋁ B hyp (simplification)
4. E’ 1,3 Modus Tollens
5. B 2,4 Disjunctive syllogism

8. Another solution 1. E → Q hyp (simplification)
2. E ⋁ B hyp (simplification)
3. Q’ hyp (simplification)
4. E’⋁ Q 1 implication
5. Q ⋁ E’ 4 commutative
6. E’ ,5 Disjunctive syllogism
7. B ,6 Disjunctive syllogism

9. Still another solution
1. E → Q hyp (simplification)
2. E ⋁ B hyp (simplification)
3. Q’ hyp (simplification)
4. Q’→ E’ contraposition
5. E’ ,3 Modus Ponens
6. (E’)’ ⋁ B 2 double negation
7. E’→ B 6 implication
8 B ,5 Modus Ponens

10. And yet another one 1. E → Q hyp (simplification)
2. E ⋁ B hyp (simplification)
3. Q’ hyp (simplification)
4. (E’)’ ⋁ B double negation
5. E’→ B 4 implication
6. E’ ⋁ Q 1 implication
7. (E’ ⋁ Q) ⋀ Q’ ,3 conjunction
8. Q’ ⋀ (E’ ⋁ Q) commutative
9. (Q’ ⋀ E’) ⋁ (Q’ ⋀ Q) 8 distributive
10. (E’ ⋀ Q’) ⋁ complement property
11. (E’ ⋀ Q’) identity property
12. E’ simplification
13. B 5,12 Modus Ponens

13. Definition: Compound proposition
“a proposition constructed by combining propositions using logical operators”
Conjunction – and
Disjunction - or

14. Definition: dual of a compound proposition
The dual of a compound proposition that contains only the logical operators ∨,,¬, is the proposition obtained by replacing each ∨ by , each  by ∨, each T by F and each F by T.

15. Definition: NAND
The proposition p NAND q is true when either p or q or both are false; and it is false when both p and q are true.

16. Definition: NOR
The proposition p NOR q is true when both p and q are false, and it is false otherwise.

17. Page 27, problem 34
Find a compound proposition involving the propositions p,q and r that is true when p and q are true and r is false, but false otherwise.

18. Page 27, problem 35
Find a compound proposition involving the propositions p, q, and r that is true when exactly two of p, q, and r are true and is false otherwise.
Hint: form a disjunction of conjunctions. Include a conjunction for each combination of values for which the proposition is true. Each conjunction should include each of the three propositions or their negation.

19. Modular Arithmetic a.k.a. clock arithmetic
If a and b are integers and m is a positive integer, then we way that a is congruent to b modulo m if m divides a-b.
We use the notation a≡b (mod m) to indicate that a is congruent to b modulo m
If they are not congruent, we write
a≢ b (mod m)

20. Why do we care in c.s.? Applications:
Stripping digits from a base 10 number
Check digits on credit cards
Hash codes for storing information
Generation of pseudo-random numbers
Cryptology
Check digits on bar codes
Enable error detection