Ma/CS 6c Assignment #1 Due Wednesday, April 7 at 1 p.m. (40%) 1 . (a) Prove the correctness of the following algorithm for recognizing when a given string S is a P -w: If S = s1 s2 . . . sn , compute w(sn ), w(sn ) + w(sn1 ), . . . , w(sn ) + w(sn1 ) + +
Introduction To Discrete Math
Instructor: Mohamed Omar
Assignment 7
Due: Tuesday Nov. 20, 2012. 12:00pm
Math 6a
1. (a) Let G be a graph on n vertices. Prove that G is connected if and only if all the
non-diagonal entries of
A(G) + A2 (G) + + An (G)
are no
Introduction To Discrete Math
Instructor: Mohamed Omar
Assignment 5
Due: Monday Nov. 5, 2012. 11:59pm
Math 6a
1. (No Collaboration)
(a) What is the order of the permutation (13)(486)(79) in S9 ? How many permutations have the same cycle type as it?
(b) De
Introduction To Discrete Math
Instructor: Mohamed Omar
Assignment 6
Due: Tues Nov. 13, 2012. 12:00pm
Math 6a
1. Determine the number of ways to color the faces of the following solid gure, where
two colorings are equivalent if one can be obtained by anoth
Introduction To Discrete Math
Instructor: Mohamed Omar
Assignment 4
Due: Monday Oct. 29, 2012. 11:59pm
Math 6a
For all problems on assignments, you are allowed to use the textbook, class notes, and other
book references. All problems must be written on yo
Introduction To Discrete Math
Instructor: Mohamed Omar
Assignment 3
Due: Monday Oct. 22, 2012. 11:59pm
Math 6a
For all problems on assignments, you are allowed to use the textbook, class notes, and other
book references. All problems must be written on yo
Introduction To Discrete Math
Instructor: Mohamed Omar
Assignment 1
Due: Monday Oct. 8, 2012. 11:59pm
Math 6a
1. (No Collaboration)
(a) Using the Euclidean algorithm, determine gcd(248399, 282041).
(b) A plane has capacity 15, 921 pounds. A company wants
Introduction To Discrete Math
Instructor: Mohamed Omar
Assignment 2
Due: Monday Oct. 15, 2012. 11:59pm
Math 6a
For all problems on assignments, you are allowed to use the textbook, class notes, and other
book references. All problems must be written on yo
Ma/CS 6c Assignment #9 Due Thursday, June 2 at 1 p.m. (30%) 1. Consider Turing machines on the alphabet cfw_1, . Let Tn be the set of Turing machines on this alphabet that have at most n states. Clearly Tn is nite. For each T M M on cfw_1, , let PM be den
Ma/CS 6c Assignment #7 Due Thursday, May 27 at 1 p.m. (20%) 1. Call a set X N eventually periodic if there is n0 N and p N, p 1 (a period) such that for all n n0 , n X i n + p X. Show that every eventually periodic set is rst-order denable in the structur
Ma/CS 6c Assignment #6 Due Thursday, May 13 at 1 p.m. (20%) 1. Prove that a proper initial segment of a formula in rst-order logic is not a formula. (10%) 2. (a) Write a sentence A6 in the language with no non-logical symbols (but recall that = is availab
Ma/CS 6c Assignment #5 Due Thursday, May 6 at 1 p.m. IN ALL PROBLEMS BELOW YOU CANNOT USE THEOREM 1.11.5 IN THE NOTES (since the exercises below form parts of the proof of that theorem). ALSO FORMULA OR WFF MEANS WELL-FORMED FORMULA IN PROPOSITIONAL LOGIC
Ma/CS 6c Assignment #4 Due Thursday, April 29 at 1 p.m. (35%) 1. Study the handout describing an algorithm for transforming a w A to a cnf formula B , so that A is satisable i B is satisable, and prove the correctness of this algorithm. (30%) 2. (i) Using
Ma/CS 6c Assignment #3 Due Wednesday, April 21 at 1 p.m. (35%) 1. (i) Prove that cfw_, , cfw_, are not complete. (ii) Prove that |, are the only complete binary connectives. (iii) Prove that, If. . . , then. . . , else. . . is not complete, but if we add
Ma/CS 6c Assignment #2 Due Wednesday, April 14, at 1 p.m. (30%) 1. (i) Let A be a w and assume that the only connectives appearing in A are among , , (i.e., , dont appear). Let A be obtained from A by replacing each propositional variable p appearing in A
Introduction To Discrete Math
Instructor: Mohamed Omar
Assignment 8
Due: Friday Dec. 7, 2012. 12:00pm
Math 6a
1. Let A be a nite set with subsets A1 , . . . , An and let d1 , d2 , . . . , dn be positive integers.
Show that there are disjoint subsets Dk Ak