MATH 520	INTERMEDIATE LOGIC	Line no: 43172
Instructor: Gábor Hetyei		Last update: April 29, 1999

Homework assignments

Disclaimer: The information below comes with no warranty. If, due to typocraphical error, there is a discrepancy between the exercise numbers announced in class and the numbers below, or this page is not completely up to date, the required homework consists of those exercises which were announced in class. Check for the time of last update above. If, by my mistake, a wrong exercise number shows up below, I will allow you extra time to hand in the exercise whose number was announced in class. If, however, exercise numbers are missing because this page is not up to date, it is your responsability to contact me before the due date. (No extra time will be allowed in that case.) This page is up to date if the last update happened after the last class before the next due date.

How to read this list:: E 2.1 means Exercise 1 from Chapter 2.
P 2.2 means Problem 2 from Chapter 2.
If there is a question in parantheses after the number, you need to hand in only the answer to that question.

Date due Exercises:

January 26, 1999 E 2.1, P 2.2, P 2.5, P 2.6, P2.7

February 2, 1999 P 2.12, P 2.13, P 2.15, P 2.24,
P 3.4 (what is the maximum number of sentences that can be made true ?), P 3.6,
P 3.8 (negative normal form), P 3.10 (negative normal form)

February 9, 1999 P 3.19, P 3.22 (1-3 only), P 3.27, P 3.34, P 3.37

February 16, 1999 P 3.40/1-3 (4. was done in class), P 3.42, P 3.46 (formal proofs only, -mirroring the informal proofs shown in class), P 3.49 (you may cite any previous proof from the book or the lecture), P 3.51 (do not hand in, will not be graded), P 3.52, P 3.56, P 3.59

February 23, 1999 P 4.3, P 4.4 (you may want to do P 4.6 to check your answers. Only the following sentences will be graded: 3,5,8,9,11,12,13,14,18,19).
P 4.18 (informal proof of what is true. If you think that the answer to any of the question is no, give a counterexample, i.e., assign truth values to the atomic sentences such that the premises hold without the conclusion being true), P 4.16/2,3 (formal proofs only, but if you are not sure of yourself, hand in an informal proof too for partial credit), P 4.24/1 (formal proof), P 4.26 (a statement like "Tet(a) & Cube(a) " is considered a contradiction).

March 2, 1999 4.29/1, 5.1, 5.2, 5.3, 4.20/4 (you may cite theorems from the list I handed out), 5.10, 5.19.

March 9, 1999 P 5.8 (hand in the solutions only, no justification necessary), P 5.9 (hand in the sentences), P 5.26. (informal proof suffices), 5.41/1,3 (hint: in 1. you need a big existential elim with a negation intro inside, in 3. you need a big universal intro with a negation intro inside), P 5.36.

March 30, 1999 P 6.9: tell in one sentence what is wrong with the proposed proof. (Hint: What kind of "arbitrary object" was b?)
P 6.14 (Hint: create first three objects based on sentences 1,5,6,9, then place them using sentences 7,8,9,10. Note that in sentence 9, Between(x,y,z) is false when x=z.)
P 6.16 (give me only the sentences), P. 6.18, P 6.19

April 6, 1999 P 6.28, P 6.38 (translate sentences into everyday English and give informal proof), P 6.46/3,4, P 6.47, P 6.49/2-4, P 6.51. Bonus homework: P 6.48 (give informal proof and hand it in on a separate sheet. Same deadline.)

April 13, 1999 P 7.3, 7.17/3,5, 7.19, 7.8/1,3,4, 7.9, 7.10, 7.27, 7.28.

April 20, 1999 P 8.2/1, P 8.6, P 8.11/1, P 8.13, P 8.23 (give list description), P 8.26.

April 27, 1999 P 9.14, P 9.7

Assume f(0)=3, f(1)=6, f(2)=12, and the recursion formula f(n+2)=4f(n-1)+4f(n)-f(n+1).
Prove by induction that f(n)=3*2^n.
Note: By mistake, an incorrect version of this problem was given in class. If I confused you, you will have an extra week to do this one.
Assume f(0)=-1, f(1)=1, and the recursion formula f(n+1)=2f(n)+3f(n-1).
Prove by induction that f(n) is less than 2.

Prove by induction that 1/(1*2)+1/(2*3)+...+1/(n*n+1)=1-1/(n+1).
Bonus homework: P 9.15. (No deadline.)

May 4, 1999 Assume f(0)=3, f(1)=6, f(2)=12, and the recursion formula f(n+2)=4f(n-1)+4f(n)-f(n+1).
Prove by induction that f(n)=3*2^n.
Note: Last week an incorrect version of this problem was announced. If you were able to correct the false statement, and to prove the corrected version, you do not have to redo this problem. Otherwise make sure that you hand it in on a separate sheet.
Compute the greatest common divisor of 180 and 350, using the Euclidean algorithm.
P 10.3/1-4, P 10.10 (Use De Morgan's rules to convert the sentence to CNF first).

Date due	Exercises:
January 26, 1999	E 2.1, P 2.2, P 2.5, P 2.6, P2.7
February 2, 1999	P 2.12, P 2.13, P 2.15, P 2.24, P 3.4 (what is the maximum number of sentences that can be made true ?), P 3.6, P 3.8 (negative normal form), P 3.10 (negative normal form)
February 9, 1999	P 3.19, P 3.22 (1-3 only), P 3.27, P 3.34, P 3.37
February 16, 1999	P 3.40/1-3 (4. was done in class), P 3.42, P 3.46 (formal proofs only, -mirroring the informal proofs shown in class), P 3.49 (you may cite any previous proof from the book or the lecture), P 3.51 (do not hand in, will not be graded), P 3.52, P 3.56, P 3.59
February 23, 1999	P 4.3, P 4.4 (you may want to do P 4.6 to check your answers. Only the following sentences will be graded: 3,5,8,9,11,12,13,14,18,19). P 4.18 (informal proof of what is true. If you think that the answer to any of the question is no, give a counterexample, i.e., assign truth values to the atomic sentences such that the premises hold without the conclusion being true), P 4.16/2,3 (formal proofs only, but if you are not sure of yourself, hand in an informal proof too for partial credit), P 4.24/1 (formal proof), P 4.26 (a statement like "Tet(a) & Cube(a) " is considered a contradiction).
March 2, 1999	4.29/1, 5.1, 5.2, 5.3, 4.20/4 (you may cite theorems from the list I handed out), 5.10, 5.19.
March 9, 1999	P 5.8 (hand in the solutions only, no justification necessary), P 5.9 (hand in the sentences), P 5.26. (informal proof suffices), 5.41/1,3 (hint: in 1. you need a big existential elim with a negation intro inside, in 3. you need a big universal intro with a negation intro inside), P 5.36.
March 30, 1999	P 6.9: tell in one sentence what is wrong with the proposed proof. (Hint: What kind of "arbitrary object" was b?) P 6.14 (Hint: create first three objects based on sentences 1,5,6,9, then place them using sentences 7,8,9,10. Note that in sentence 9, Between(x,y,z) is false when x=z.) P 6.16 (give me only the sentences), P. 6.18, P 6.19
April 6, 1999	P 6.28, P 6.38 (translate sentences into everyday English and give informal proof), P 6.46/3,4, P 6.47, P 6.49/2-4, P 6.51. Bonus homework: P 6.48 (give informal proof and hand it in on a separate sheet. Same deadline.)
April 13, 1999	P 7.3, 7.17/3,5, 7.19, 7.8/1,3,4, 7.9, 7.10, 7.27, 7.28.
April 20, 1999	P 8.2/1, P 8.6, P 8.11/1, P 8.13, P 8.23 (give list description), P 8.26.
April 27, 1999	P 9.14, P 9.7 Assume f(0)=3, f(1)=6, f(2)=12, and the recursion formula f(n+2)=4f(n-1)+4f(n)-f(n+1). Prove by induction that f(n)=32^n. Note:* By mistake, an incorrect version of this problem was given in class. If I confused you, you will have an extra week to do this one. Assume f(0)=-1, f(1)=1, and the recursion formula f(n+1)=2f(n)+3f(n-1). Prove by induction that f(n) is less than 2. Prove by induction that 1/(12)+1/(23)+...+1/(nn+1)=1-1/(n+1). Bonus homework:* P 9.15. (No deadline.)
May 4, 1999	Assume f(0)=3, f(1)=6, f(2)=12, and the recursion formula f(n+2)=4f(n-1)+4f(n)-f(n+1). Prove by induction that f(n)=32^n. Note:* Last week an incorrect version of this problem was announced. If you were able to correct the false statement, and to prove the corrected version, you do not have to redo this problem. Otherwise make sure that you hand it in on a separate sheet. Compute the greatest common divisor of 180 and 350, using the Euclidean algorithm. P 10.3/1-4, P 10.10 (Use De Morgan's rules to convert the sentence to CNF first).