Math 6a - Solution Set 7
Problem # 1 Liu 5.15 Let T1 and T2 be two spanning trees of a connected graph G. Let a be an edge that is in T1 but not T2 . Prove that there is an edge b in T2 but not T1 such that both (T1 - cfw_a) cfw_b and (T2 - cfw_a) cfw_b a
MATH 6A FALL 2011 HOMEWORK 4 SOLUTIONS
1. Solution:
a. A(x) =
32n x2n =
n=0
1
19x2 .
b. Note that if B (x) =
bn xn , C (x) =
n=0
cn xn , then the product of power
n=0
series B (x)C (x) would have coecient of xn be
n
bi cni .
i=0
5
Thus A(x) = B (x)C (x) a
MATH 6A FALL 2011 HOMEWORK 3 SOLUTIONS
1. Solution:
a. First we consider triples. For any xed number, there are 5 possible triples,
3
because we could have triples of diering colors. Thus we have 8 5 triples. Now
3
we have pairs remaining. Of the pairs, t
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
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
MATH 6A FALL 2011 HOMEWORK 1 SOLUTIONS
1. Solution:
a. The algorithm is the so called repeated squaring method.
To compute ab (modc).
First, write out the binary presentation of b, its just divide b by 2 repeatedly.
i
i+1
Now, suppose we already know a2 (
MATH 6A FALL 2011 HOMEWORK 3 SOLUTIONS
1. Solution:
a. Given the equation x1 + x2 + x3 = 21 where xi = 2ki 1 are odd positive
integers, we have a new equation:
k1 + k2 + k3 = 12
where ki are positive integers.
Its then equivalents to the combinatorial pro
MATH 6A FALL 2011 HOMEWORK 5 SOLUTIONS
1. Solution:
a. The order of (13)(486)(79) is 6, and its type in S9 is 12 22 31 .
The number of permutations of this type is (12 29! 1 )2! = 15120(For any arrange23
ment of cfw_1, 2, , 9, we pick up the cycles in cfw
MATH 6A FALL 2011 HOMEWORK 6 SOLUTIONS
1. Solution:
Lemma 1. Assume G has an action on A, then we have an induced action of G
on B A := cfw_f : A B via (g.f )(x) = f (g 1 x). And the number of orbits of G
acts on B A is
1
ck (G)|B |k
|G|
k=1
where ck (G)
Fall 2011 Math/CS 6A FINAL EXAM
Due: 11:59 p.m. on December 9th, 2011
Return Exam To: Math/CS 6A Assignment Drop Box
The exam time period begins when you open your exam. You must complete the exam
24 hours after that time. You may not communicate with any
MATH 6A FALL 2011 HOMEWORK 5 SOLUTIONS
1. Solution:
a. The order of (13)(486)(79) is lcm(2, 3) = 6, and its type in S9 is 12 22 31 .
Now consider the map
S9 S9 : ( (0) (1)( (2) (3)( (4) (5) (6).
This clearly maps S9 onto all elements of S9 of type 12 22 3
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
Math 6a - Midterm Solutions Problem # 1 A commutative ring is a set B with two binary operations + and called addition and multiplication and two constants 0 and 1 such that: (a) Addition is associative and commutative (b) For all b in B, b + 0 = b = 0 +
Math 6a - Solution Set 1 Problem # 1: Liu 1.30 (d) Determine and prove a general formula that includes the results in (a), (b) and (c) as special cases. Solution. The general formula, which can easily be inferred by observing the formulae in the previous
Math 6a - Solution Set 2 Problem # 1 Is Liu, Example 2.14 correct? Solution. In fact, it is. Problem # 2: Liu 2.32 Find the number of permutations of the letters a, b, c, d, e, f, g so that neither the pattern beg nor the pattern cad appears. (Hint: Use t
Math 6a - Solution Set 3 Problem # 1 Story problem 1 "My question for you, dear Carib, is this: Can you find another truth-functional connective that is associative? That is, is there some well defined way of taking two truth values, p and q, and producin
Math 6a - Solution Set 4 Problem # 1 Liu 4.10 A set of vertices in an undirected graph is said to be a dominating set if every vertex not in the set is adjacent to one or more vertices in the set. A minimal dominating set is a dominating set such that no
Math 6a - Solution Set 5 Problem # 1 Show that S(n, 2) = 2n-1 - 1 and S(n, 3) = 1 (3n-1 - 2n + 1) for n 1. 2 Solution. Note that S(n, 2) counts the number of partitions of an n element set S into 2 non-empty subsets. So if we call the sets A and B, each e
Math 6a - Solution Set 6 Problem # 1 Write a paragraph on dominance in graphs, fire fighting, and the spread of computer viruses, with reference to the work of Hartnell. Write a paragraph on what binary search trees have to do with port security, with ref
Math 6a - Solution Set 8 Problem # 1 Liu 4.2 (a) Three married couples on a journey come to a river where they find a boat which cannot carry more than two persons at a time. The crossing of the river is complicated by the fact that the husbands are all v
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
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
Ma/CS 6a Brief solutions to selected problems, Set 2
1. Assume that the three numbers 6k + 1, 12k + 1, and 18k + 1 are all prime numbers,
and let
n = (6k + 1)(12k + 1)(18k + 1).
Let a and n be coprime. Then a is coprime to the three prime factors of n. We