1. How many four letter words (including nonsensical words) are there that
do not have two consecutive letters the same? How many are there that
do have two consecutive letters the same ?
We can choose the rst letter in 26 ways. Having chosen the rst lett
1. Write out the following sets by listing their elements between curly braces.
For each set, give its cardinality.
(a) fx 2 Z : x j 21g
The set is f 21; 7; 3; 1; 1; 3; 7; 21g. Its cardinality is 8.
(b) fx : x f1; 2; 3; 4g and jxj 2g
The set is f; f1g ; f
1. Give an example of a relation that is symmetric but not transitive.
The smallest is f(1; 2); (2; 1)g.
2. Give an example of a relation that is transitive but not symmetric.
The smallest is f(1; 2)g.
3. How many relations on the set f1; 2; 3g are both r
1. How many di erent anagrams (including nonsensical words) can be made
from the word TALLAHASSEE?
= 831; 600
2. List all the partitions of the set f1; 2; 3; 4g that have three parts.
ff1; 2g ; f3g ; f4gg
ff1; 3g ; f2g ; f4gg
ff1; 4g ; f2g ;
1. How many ways can you arrange the seven characters a, b, c, d, 1, 2, 3 in
a row so that
(a) the letters are in alphabetical order ?
You can place the numbers anywhere in the seven positions of the
row, so there are 7 6 5 = 210 ways to place the numbers
1. List all the size-three multisets whose elements are selected from the set
fa; b; cg. Then give the stars-and-bars representation for each of the multisets you listed. Check that the number of multisets you listed is the same
as the number you get when
1. Show by mathematical induction that 1 + 3 + 32 +
every positive integer n.
+ 3n =
The base case is when n = 1, in which case the equation is 1 + 3 =
which is true.
For the induction step, we assume that the equation holds for n =
1. Let the sequence a0 ; a1 ; a2 ; : : : be dened by a0 = 0 and an = an
n > 0. Show that an = n (n + 1) =2 for all positive integers n.
+ n for
We show that the formula is true by induction on n. The base case is
n = 0. We have a0 = 0 and 0 (0 + 1) =2 =
1. Find integers q and r so that 185 = 22q + r and 0
( 185) mod 22 and ( 185) div 22?
r < 22. What are
One way to do this is to compute 185=22 = 8:4091 on a hand calculator.
That tells us that q = 9, the integer just to the left of 8:4091. So
r = 185 22q
1. A graph has 7 vertices, with degrees 0, 1, 2, 2, 3, 4, and 4.
(a) How many edges does the graph have ?
(b) Draw an example of such a graph.
One such graph is this
2. Let G be a graph with 10 vertices and 15 edges.
1. Let Kn be the complete graph on n vertices (each two vertices are adjacent ), and Km;n the complete bipartite graph : V = X [ Y where X \ Y = ;,
jX j = m, jY j = n, and E = ffx; y g : x 2 X and y 2 Y g.
(a) How many vertices and edges does K7 have ?
1. In how many ways can we arrange a standard deck of 52 cards so
that all cards of a given suit appear contiguously (for example, the rst
thirteen cards are spades, the next thirteen are diamonds, then all the
hearts, and then all the clubs)?
Each suit c
In Question 1 you had to remember in each part to give the cardinality
of the set.
In Question 1 part (a), the expression x j 21 means that 21 is divisible by
x. That is, 21 is a multiple of x. That is, 21 = mx for some integer m. So,
for example, 21 j 21
1. We noticed that a graph with more than two vertices of odd degree
cannot have an Eulerian trial, but connected graphs with zero or two
vertices of odd degree do have Eulerian trails. The missing case is
connected graphs with exactly one vert
1. Let G be the graph in the gure.
(a) How many dierent paths are there from a to b?
Each path has to go through that point w in the middle (the waist
of the graph). There are 5 paths from a to w and 3 paths from w
to b, so there are 15 = 3 5 p
Annotated answers to the January 22 assignment
1. (Page 57)
(a) f0; 3; 6; 9g. Don forget that 3j0 because 0 = 3 0.
(b) f2g. Dierent people dene primein dierent ways. The book
denition is on page 3. You can nd that out by looking up
primein the index.
Answers to Homework
31 January 2008
13.1 Relations on A = f1; 2; 3; 4; 5g.
(a) f(1; 1); (2; 2); (3; 3); (4; 4); (5; 5)g is the equality relation. It is re
exive, symmetric, antisymetric and transitive.
(b) f(1; 2); (2; 3); (3; 4); (4; 5)g is the relation
Answers to Homework
7 February 2008
1. Find all possible partitions of the set f1; 2; 3g.
Classify them by how many sets are in the partition.
One partition into one set ff1; 2; 3gg.
Three partitions into two sets ff1g ; f2; 3gg, ff2g ; f1; 3gg
Answers to Homework
14 February 2008
4. How many di erent anagrams (including nonsensical words) can be made
from FACETIOUSLY if we require that all six vowels must remain in
alphabetical order (but not necessarily contiguous with each other).
Answers to Homework
21 February 2008
1. Evaluate 3 and 2 by explicitly listing all possible multisets of the
appropriate size. Check that your answers agree with the formula in Theorem 17.8.
the multisets are
h1; 1i , h2; 2i , h3; 3
2. Show that
n(n + 1)
for all positive integers n.
The basis step is to verify that the equation holds for n = 1. In this
case the left-hand side is 1 and the right-hand side is 1(1 + 1)=2 = 1.
The induction step is to show that if
5. Show that Fn > 1:6n once n is big enough.
The basis step here is to nd a place where the inequality is true
for two consecutive values of n. That because the induction can go
from n 1 to n, we generate Fn = Fn 2 + Fn 1 from the two prece
1. For the pairs of integer a, b given below, nd the integers q and r such
that a = qb + r and 0 r < b
(a) a = 100 and b = 3.
q = 33 and r = 1.
(b) a =
100 and b = 3.
34 and r = 2.
(c) a = 99 and b = 3.
q = 33 and r = 0.
(d) a =
99 and b
1. Working in Z10
(a) 3 + 3 = 6
(b) 6 + 6 = 2
(c) 7 + 3 = 0
(d) 9 + 8 = 7
(e) 12 + 4 =? Technically, according to the denition of Zn on page
310, the number 12 is not in Z10 , so this addition is not dened.
Personally, in the context of Z10 , I
SAMPLE TEST # 1
1) Construct the Truth Table for each of the following Boolean expressions:
Are they equivalent expressions? Are they tautologies? Contradictions?