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

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

Numerade Educator

02:00

Problem 3

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

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

Numerade Educator

01:34

Prove that no two integers in $Z_{n}$, arithmetic mod $n$, have the same additive inverse. Conclude from the pigeonhole principle that
$$
\{-0,-1,-2, \ldots,-(n-1)\}=\{0,1,2, \ldots, n-1\}
$$
(Remember that $-a$ is the integer which, when added to $a$ in $Z_{n}$, gives $0 .$ )

Clarissa Noh

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

Numerade Educator

00:42

Problem 9

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

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

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

Numerade Educator

03:00

Problem 12

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

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

Numerade Educator

08:00

Problem 14

Use the algorithm in Section $10.1$ to find the GCD of the following pairs of integers:
(i) 12 and 31
(ii) 24 and 82
(iii) 26 and 97
(iv) 186 and 334
(v) 423 and 618

Bryan Lynn

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

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

Numerade Educator

01:35

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:

Sherrie Fenner

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

Indiana University Bloomington

01:24

Problem 19

\begin{array}{l}
\text { Does there exist a BIBD whose parameters satisfy } b=20, v=18 \text { , }\\
k=9, \text { and } r=10 ?
\end{array}

Whitney Massock

Numerade Educator

00:35

Problem 20

Let $\mathcal{B}$ be a BIBD with parameters $b, v, k, r, \lambda$ whose set of varieties is $X=\left\{x_{1}, x_{2}, \ldots, x_{0}\right\}$ 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 $B^{c}$ be the collection of subsets $\overline{B_{1}}, \overline{B_{2}}, \ldots, \overline{B_{b}}$ of $X$. Prove that $\mathcal{B}^{c}$ is a block design with parameters
$$
b^{\prime}=b, v^{\prime}=v, k^{\prime}=v-k, r^{\prime}=b-r, \lambda^{\prime}=b-2 r+\lambda_{1}
$$
provided that we have $b-2 r+\lambda>0$. The BIBD $\mathcal{B}^{c}$ is called the complementary design of $\mathcal{B}$.

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

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

Numerade Educator

00:48

Problem 23

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

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

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

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 of 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.

California State Polytechnic University, Pomona

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Problem 27

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

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

Numerade Educator

02:57

Problem 29

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

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

Numerade Educator

01:22

Problem 31

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

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

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

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

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:
(i) If $a=1$ or 5, then $v$ has remainder 1 or 3 when divided by $6 .$
(ii) If $a=2$ or 4, then $v$ has remainder 0 or 1 when divided by $3 .$
(iii) If $a=3$, then $v$ is odd.

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 $\left.Z_{13}\right)$.
(i) Each of the integers $1,3,4,9,10,12$ occurs exactly once as a difference of two integers in $B_{1}=\{0,1,4\}$.
(ii) Each of the integers $2,5,6,7,8,11$ occurs exactly once as a difference of two integers in $B_{2}=\{0,2,7\}$.
(iii) 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

Numerade Educator

03:34

Problem 37

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

Tawana Stiff

Numerade Educator

01:13

Problem 38

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

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 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

Numerade Educator

36:47

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 Theoro

Oswaldo Jiménez

Numerade Educator

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Problem 41

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

Alison Rodriguez

Numerade Educator

12:49

Problem 42

Construct 4 MOLS of order 5 .

Dr. Satish Ingale

Numerade Educator

03:02

Problem 43

Construct 3 MOLS of order 7 .

Ruby P

Numerade Educator

00:29

Problem 44

Construct 2 MOLS of order 9 .

Matt Gibson

Numerade Educator

01:43

Problem 45

Construct 2 MOLS of order 15 .

Preeti Kumari

Numerade Educator

01:25

Problem 46

Construct 2 MOLS of order 8 .

Kevin Chimex

Numerade Educator

10:14

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 's in $A$ as rooks of color red, the l's as rooks of color white, the 2's 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 two rooks in the same set having the same color.

Paul A.

California State Polytechnic University, Pomona

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

Numerade Educator

00:38

Problem 49

First construct 4 MOLS of order 5, and then construct the resolvable BIBD corresponding to them as given in Theorem $10.4 .10 .$

Amrita Bhasin

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

Numerade Educator

01:24

Problem 51

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

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

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

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

Numerade Educator

11:45

Problem 55

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

Sirat Shah

Numerade Educator

01:07

Problem 56

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

Jake Zanazzi

Numerade Educator

04:41

Problem 57

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

Sarvesh Somasundaram

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 $a_{j i}$ 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

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 $0,1,2, \ldots, n-1 .$
(i) Construct an example of an idempotent Latin square of order $5 .$
(ii) Construct an example of a symmetric, idempotent Latin square of order $5 .$

Manisha Sarker

Numerade Educator

01:28

Problem 60

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

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
$$
a_{i j}=(m+1) \times(i+j)(\text { arithmetic mod } n)
$$
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 $\left.j .\right]$

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

Numerade Educator