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Introductory Combinatorics

Richard A. Brualdi

Chapter 10

Combinatorial Designs - all with Video Answers

Educators

Chapter Questions

04:05

Problem 1

Compute the addition table and the multiplication table for the integers mod 4 .

Anurag Kumar
Anurag Kumar
Numerade Educator
01:56

Problem 2

Compute the subtraction table for the integers mod 4. How does it compare with the addition table computed in Exercise $1 ?$

Nick Johnson
Nick Johnson
Numerade Educator
02:00

Problem 3

Compute the addition table and the multiplication table for the integers mod 5 .

James Chok
James Chok
Numerade Educator
00:12

Problem 4

Compute the subtraction table of the integers mod 5 . How does it compare with the addition table computed in Exercise 3 ?

Amy Jiang
Amy Jiang
Numerade Educator
02:24

Problem 5

Prove that no two integers in $Z_{n}$, arithmetic mod $n$, have the same additive inverse. Conclude from the pigeonhole principle that
$$\{-0,-1,-2, \ldots,-(n-1)\}=\{0,1,2, \ldots, n-1\} .$$

Kumar Vaibhav
Kumar Vaibhav
Numerade Educator
01:54

Problem 6

Prove that the columns of the subtraction table of $Z_{n}$ are a rearrangement of the columns of the addition table of $Z_{n}$ (cf. Exercises 2 and 4 ).

Raushan Kumar
Raushan Kumar
Numerade Educator
03:19

Problem 7

Compute the addition table and multiplication table for the integers mod 6 .

James Chok
James Chok
Numerade Educator
00:37

Problem 8

Determine the additive inverses of the integers in $Z_{8}$, with arithmetic mod 8 .

Zach Steedman
Zach Steedman
Numerade Educator
00:42

Problem 9

Determine the additive inverses of $3,7,8$, and 19 in the integers mod 20 .

Kellyn Toombs
Kellyn Toombs
Numerade Educator
05:05

Problem 10

Determine which integers in $Z_{12}$ have multiplicative inverses, and find the multiplicative inverses when they exist.

Ruby P
Ruby P
Numerade Educator
00:12

Problem 11

For each of the following integers in $Z_{24}$, determine the multiplicative inverse if a multiplicative inverse exists:
$$4,9,11,15,17,23 .$$

Christine Girgus
Christine Girgus
Numerade Educator
03:00

Problem 12

Prove that $n-1$ always has a multiplicative inverse in $Z_{n},(n \geq 2)$.

Mitchell Riley
Mitchell Riley
Numerade Educator
00:46

Problem 13

Let $n=2 m+1$ be an odd integer with $m \geq 2$. Prove that the multiplicative inverse of $m+1$ in $Z_{n}$ is 2 .

Jay Patel
Jay Patel
Numerade Educator
11:58

Problem 14

Use the algorithm in Section $10.1$ to find the GCD of the following pairs of integers:
(a) 12 and 31
(b) 24 and 82
(c) 26 and 97
(d) 186 and 334
(e) 423 and 618

Raphael Tinoco
Raphael Tinoco
Numerade Educator
08:04

Problem 15

. For each of the pairs of integers in Exercise 14 , let $m$ denote the first integer and let $n$ denote the second integer of the pair. When it exists, determine the multiplicative inverse of $m$ in $Z_{n}$.

Trang Hoang
Trang Hoang
Numerade Educator
07:52

Problem 16

Apply the algorithm for the GCD in Section $10.1$ to 15 and 46 , and then use the results to determine the multiplicative inverse of 15 in $Z_{46}$.

Raphael Tinoco
Raphael Tinoco
Numerade Educator
01:37

Problem 17

Start with the field $Z_{2}$ and show that $x^{3}+x+1$ cannot be factored in a nontrivial way (into polynomials with coefficients in $Z_{2}$ ), and then use this polynomial to construct a field with $2^{3}=8$ elements. Let $i$ be the root of this polynomial adjoined to $Z_{2}$, and then do the following computations:
(a) $(1+i)+\left(1+i+i^{2}\right)$
(b) $\left(1+i^{2}\right)+\left(1+i^{2}\right)$
(c) $i^{-1}$
(d) $i^{2} \times\left(1+i+i^{2}\right)$
(e) $(1+i)\left(1+i+i^{2}\right)$
(f) $(1+i)^{-1}$

AG
Ankit Gupta
Numerade Educator
01:13

Problem 18

Does there exist a BIBD with parameters $b=10, v=8 r=5$, and $k=4$ ?

Jonathan Kerby-White
Jonathan Kerby-White
Indiana University Bloomington
01:24

Problem 19

Does there exist a BIBD whose parameters satisfy $b=20, v=18, k=9$, and $r=10 ?$

Whitney Massock
Whitney Massock
Numerade Educator
00:35

Problem 20

Let $B$ be a BIBD with parameters $b, v, k, r, \lambda$ whose set of varieties is $X=$ $\left\{x_{1}, x_{2}, \ldots, x_{v}\right\}$ and whose blocks are $B_{1}, B_{2}, \ldots, B_{b} .$ For each block $B_{i}$, let $\overline{B_{i}}$ denote the set of varieties which do not belong to $B_{i} .$ Let $\mathcal{B}^{c}$ be the collection of subsets $\overline{B_{1}}, \overline{B_{2}}, \ldots, \overline{B_{b}}$ of $X$. Prove that $\mathcal{B}^{\mathrm{c}}$ is a block design with parameters
$$b^{\prime}=b, v^{\prime}=v, k^{\prime}=v-k, r^{\prime}=b-r, \lambda^{\prime}=b-2 r+\lambda$$
provided that we have $b-2 r+\lambda>0 .$ The $\mathrm{BIBD} B^{c}$ is called the complementary design of $\mathcal{B}$.

Mohamed Mohamed
Mohamed Mohamed
Numerade Educator
03:02

Problem 21

Determine the complementary design of the BIBD with parameters $b=v=$ $7, k=r=3, \lambda=1$ in Section $10.2 .$

Narayan Hari
Narayan Hari
Numerade Educator
03:02

Problem 22

Determine the complementary design of the BIBD with parameters $b=v=$ $16, k=r=6, \lambda=2$ given in Section $10.2$.

Narayan Hari
Narayan Hari
Numerade Educator
00:48

Problem 23

How are the incidence matrices of a BIBD and its complement related?

Elizabeth Xu
Elizabeth Xu
Numerade Educator
01:21

Problem 24

Show that a $\mathrm{BIBD}$, with $v$ varieties whose block size $k$ equals $v-1$, does not have a complementary design.

Lauren Shelton
Lauren Shelton
Numerade Educator
04:52

Problem 25

Prove that a BIBD with parameters $b, v, k, r, \lambda$ has a complementary design if and only if $2 \leq k \leq v-2$ (Cf. Exercises 20 and 24 ).

Gideon Idumah
Gideon Idumah
Numerade Educator
10:14

Problem 26

Let $B$ be a difference set in $Z_{n}$. Show that, for each integer $k$ in $Z_{n}, B+k$ is also a difference set. (This implies that we can always assume without loss ol generality that a difference set contains 0 for, if it did not, we can replace it by $B+k$, where $k$ is the additive inverse of any integer in $B .$ )

Paul A.
Paul A.
California State Polytechnic University, Pomona
View

Problem 27

Prove that $Z_{v}$ is itself a difference set, in $Z_{\mathrm{v}}$. (These are trivial difference sets.)

Hoan Nguyen
Hoan Nguyen
Numerade Educator
02:22

Problem 28

Show that $B=\{0,1,3,9\}$ is a difference set in $Z_{13}$, and use this difference set as a starter block to construct an SBIBD. Identify the parameters of the block design.

Adriano Chikande
Adriano Chikande
Numerade Educator
02:57

Problem 29

Is $B=\{0,2,5,11\}$ a difference set in $Z_{12}$ ?

Nicole C
Nicole C
Numerade Educator
02:35

Problem 30

Show that $B=\{0,2,3,4,8\}$ is a difference set in $Z_{11}$. What are the parameters of the SBIBD developed from $B ?$

Prachita Kush
Prachita Kush
Numerade Educator
01:22

Problem 31

Prove that $B=\{0,3,4,9,11\}$ is a difference set in $Z_{21}$.

AG
Ankit Gupta
Numerade Educator
00:52

Problem 32

Use Theorem $10.3 .2$ to construct a Steiner triple system of index 1 having 21 varieties.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:45

Problem 33

Let $t$ be a positive integer. Use Theorem $10.3 .2$ to prove that there exists a Steiner triple system of index 1 having $3^{t}$ varieties.

James Chok
James Chok
Numerade Educator
03:51

Problem 34

Let $t$ be a positive integer. Prove that, if there exists a Steiner triple system of index 1 having $v$ varieties, then there exists a Steiner triple system having $v^{t}$ varieties (cf. Exercise 33).

Wasim Sher
Wasim Sher
Numerade Educator
01:48

Problem 35

Assume a Steiner triple system exists with parameters $b, v, k, r, \lambda$, where $k=3$. Let $a$ be the remainder when $\lambda$ is divided by $6 .$ Use Theorem $10.3 .1$ to show the following:
(1) If $a=1$ or 5 , then $v$ has remainder 1 or 3 when divided by 6 .
(2) If $a=2$ or 4 , then $v$ has remainder 0 or 1 when divided by 3 .
(3) If $a=3$, then $v$ is odd.

Allison Knapp
Allison Knapp
Numerade Educator
05:44

Problem 36

Verify that the following three steps construct a Steiner triple system of index 1 with 13 varieties (we begin with $Z_{13}$ ).
(1) Each of the integers $1,3,4,9,10,12$ occurs exactly once as a difference of two integers in $B_{1}=\{0,1,4\}$.
(2) Each of the integers $2,5,6,7,8,11$ occurs exactly once as a difference of two integers in $B_{2}=\{0,2,7\}$.
(3) The 12 blocks developed from $B_{1}$ together with the 12 blocks developed from $B_{2}$ are the blocks of a Steiner triple system of index 1 with 13 varieties.

Megan Mcfarland
Megan Mcfarland
Numerade Educator
03:34

Problem 37

Prove that, if we interchange the rows of a Latin square in any way and inter. change the columns in any way, the result is always a Latin square.

Tawana Stiff
Tawana Stiff
Numerade Educator
01:13

Problem 38

Use the method in Theorem $10.4 .2$ with $n=6$ and $r=5$ to construct a Latin square of order 6 .

Prashansha Kaushik
Prashansha Kaushik
Numerade Educator
01:43

Problem 39

Let $n$ be a positive integer and let $r$ be a nonzero integer in $Z_{n}$ such that the $\mathrm{GCD}$ of $r$ and $n$ is not $1 .$ Prove that the array constructed using the prescription in Theorem $10.4 .2$ is not a Latin square.

Km Neeraj
Km Neeraj
Numerade Educator
02:06

Problem 40

Let $n$ be a positive integer and let $r$ and $r^{\prime}$ be distinct nonzero integers in $Z_{n}$ such that the GCD of $r$ and $n$ is 1 and the GCD of $r^{\prime}$ and $n$ is 1. Show that the Latin squares constructed by using Theorem $10.4 .2$ need not be orthogonal.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:38

Problem 41

. Use the method in Theorem $10.4 .2$ with $n=8$ and $r=3$ to construct a Latin square of order 8 .

Amy Jiang
Amy Jiang
Numerade Educator
12:49

Problem 42

. Construct four MOLS of order $5 .$

Dr. Satish Ingale
Dr. Satish Ingale
Numerade Educator
01:03

Problem 43

Construct three MOLS of order 7 .

David Collins
David Collins
Numerade Educator
00:29

Problem 44

Construct two MOLS of order $9 .$

Matt Gibson
Matt Gibson
Numerade Educator
02:14

Problem 45

Construct two MOLS of order 15 .

Ronald Prasad
Ronald Prasad
Numerade Educator
02:14

Problem 46

Construct two MOLS of order 8 .

Ronald Prasad
Ronald Prasad
Numerade Educator
01:16

Problem 47

Let $A$ be a Latin square of order $n$ for which there exists a Latin square $B$ of order $n$ such that $A$ and $B$ are orthogonal. $B$ is called an orthogonal mate of $A$. Think of the 0 in $A$ as rooks of color red, the is as rooks of color white, the 2x as rooks of color blue, and so on. Prove that there are $n$ nonattacking rooks in $A$, no two of which have the same color. Indeed, prove that the entire set of $n^{2}$ rooks can be partitioned into $n$ sets of $n$ nonattacking rooks each, with no twu rooks in the same set having the same color.

Raj Bala
Raj Bala
Numerade Educator
01:37

Problem 48

Prove that the addition table of $Z_{4}$ is a Latin square without an orthogonal mate:
(cf. Exercise 47).

Nikhil Kumar Rajpurohit
Nikhil Kumar Rajpurohit
Numerade Educator
02:01

Problem 49

First construct 4 MOLS of order 5, and then construct the resolvable $\mathrm{BIB} !$ ) corresponding to them as given in Theoren? 10.4.10.

Anas Venkitta
Anas Venkitta
Numerade Educator
07:38

Problem 50

Let $A_{1}$ and $A_{2}$ be MOLS of order $m$ and let $B_{1}$ and $B_{2}$ be MOLS of order $n$ Prove that $A_{1} \otimes B_{1}$ and $A_{2} \otimes B_{2}$ are MOLS of order $m n$.

Preeti Kumari
Preeti Kumari
Numerade Educator
01:24

Problem 51

Fill in the details in the proof of Theorem $10.4 .10$.

Nick Johnson
Nick Johnson
Numerade Educator
01:46

Problem 52

Construct a completion of the 3 -by- 6 Latin rectangle
$\left[\begin{array}{llllll}0 & 1 & 2 & 3 & 4 & 5 \\ 4 & 3 & 1 & 5 & 2 & 0 \\ 5 & 4 & 3 & 0 & 1 & 2\end{array}\right]$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:39

Problem 53

Construct a completion of the 3 -by- 7 Latin rectangle
$$\left[\begin{array}{lllllll}0 & 1 & 2 & 3 & 4 & 5 & 6 \\
2 & 3 & 0 & 6 & 5 & 4 & 1 \\1 & 4 & 6 & 0 & 2 & 3 & 5
\end{array}\right] .$$

Teresa Fuston
Teresa Fuston
Numerade Educator
01:46

Problem 54

How many 2 -by-n Latin rectangles have first row equal to
$$\begin{array}{lllll}0 & 1 & 2 & \cdots & n-1
\end{array} ?$$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
11:45

Problem 55

Construct a completion of the semi-Latin square
$$\left[\begin{array}{ccccc}
& 2 & 0 & & & 1 \\2 & 0 & & & 1 & \\
0 & & 2 & 1 & & \\& & 1 & 2 & & 0 \\
& 1 & & & 0 & 2 \\1 & & & 0 & 2 &\end{array}\right]$$

Sirat Shah
Sirat Shah
Numerade Educator
01:07

Problem 56

Construct a completion of the semi-Latin square
$$\left[\begin{array}{lllllll}0 & 2 & 1 & & & & 3 \\
2 & 0 & & 1 & & 3 & \\3 & & 0 & 2 & 1 & & \\
& 3 & 2 & 0 & & 1 & \\
& & 3 & & 0 & 2 & 1 \\1 & & & & 3 & 0 & 2 \\& 1 & & 3 & 2 & & 0\end{array}\right] .$$

Jake Zanazzi
Jake Zanazzi
Numerade Educator
02:21

Problem 57

Let $n \geq 2$ be an integer. Prove that an $(n-2)$ -by- $n$ Latin rectangle has at least two completions, and, for each $n$, find an example that has exactly two completions.

Nick Johnson
Nick Johnson
Numerade Educator
04:52

Problem 58

A Latin square $A$ of order $n$ is symmetric, provided the entry $a_{i j}$ at row $i$, column $j$ equals the entry $a_{j 1}$ at column $j$, row $i$ for all $i \neq j$. Prove that the addition table of $Z_{n}$ is a symmetric Latin square.

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator
04:10

Problem 59

A Latin square of order $n$ (based on $Z_{n}$ ) is idempotent. provided that its entries on the diagonal running from upper left to lower right are $0,1,2, \ldots, n-1$.
(1) Construct an example of an idempotent Latin square of order 5 .
(2) Construct an example of a symmetric, idempotent Latin square of order $5 .$

Manisha Sarker
Manisha Sarker
Numerade Educator
01:28

Problem 60

Prove that a symmetric, idempotent Latin square has odd order.

Srilakshmi E K
Srilakshmi E K
Numerade Educator
01:31

Problem 61

Let $n=2 m+1$, where $m$ is a positive integer. Prove that the $n$ -by-n array $A$ whose entry $a_{i j}$ in row $i$, column $j$ satisfies
$$a_{i j}=(m+1) \times(i+j) \text { (arithmetic mod } n \text { ) }$$
is a symmetric, idempotent Latin square of order $n .$ (Remark The integer $m+1$ is the multiplicative inverse of 2 in $Z_{n}$. Thus, our prescription for $a_{i j}$ is to "average" $i$ and $j$.)

Angelo Rendina
Angelo Rendina
Numerade Educator
04:52

Problem 62

Let $L$ be an $m$ -by-n Latin rectangle (based on $Z_{n}$ ) and let the entry in row $i$, column $j$ be denoted by $a_{i j}$. We define an $n$ -by-n array $B$ whose entry $b_{i j}$ in position row $i$, column $j$ satisfies
$$b_{i j}=k, \text { provided } a_{k j}=i$$
and is blank otherwise. Prove that $B$ is a semi-Latin square of order $n$ and index
$m$. In particular, if $A$ is a Latin square of order $n$, so is $B$.

Harmender Singh Yadav
Harmender Singh Yadav
Numerade Educator