'it
This work was supported by NSF grants
GU-2059 and GU-19568.
tOn leave from Punjab Agricultural University (India).
eBOOLEAN SUMS OF SETS OF CERTAIN DESIGNS*
...
Damaraju Raghavarao'
Department of Statistics
Universi ty of North Caro Una at Chape Z Hi II
Institute of Statistics Mtmeo Series No. 852
November, 1972
BOOLEAN SUMS OF SETS OF CERTAIN DESIGNS
by
Damaraju Raghavarao
1
University of Nor-t;h CaroUna at ChapeZ Hill
ABSTRACT
In this paper the author applies Boolean sum operation
on sets of certain known designs to get new series of designs.
In particular certain unsYmmetrical balanced incomplete
block designs can be constructed from sYmmetrical balanced
incomplete block designs and triangular designs can be
constructed from irreducible balanced incomplete block
designs and tactical configurations
$olutioas obtained by our
e·
some of the known designs.
ue·.:~"od
C[k, 4, 0, v].
The
are nOli.··isomorphic to
The possible generalizations
of these concepts are also considered.
1. INTRODUCT ION
A tactical configuration (or simply configuration)
is defined to be a system of subsets, of a set
k
symbols each, such that every subset of
contained in exactly 0
C[k,~,o,v]
E of
v
symbols, having
E having
~
symbols is
sets of the system (cf. Raghavarao (1971».
A tactical configuration
C[k,~,o,
balanced incomplete block (BIB) design.
v] when
~
=2
is
kno.~
as a
For a BIB design it is customary
0 by A. Let each of the v symbols occur in r subsets of
the configuration and let there be b subsets in the system. Then v,
to denote
1 On leave from Punjab Agricultural University (India).
supported by NSF grants GU-2059 and GU-19568.
This work was
2
b, r, k, and A are known as the parameters of the BIB design.
design is said to be symmetrical if
v
= b;
The BIB
otherwise unsymmetrical.
The BIB design with parameters
=
v, b
(~)
,
r
V-I)
= (k-l
'
= (v-2)
k-2
k,A
where (~) denotes the number of combinations of
symbols, is called irreducible
(i)
(ii)
(iii)
e-
r
symbols from
n
BIB design.
A triangular design is an arrangement of
sets each of size
(1.1)
v
= n(n-l)/2
symbols ih
k«v) such that
every symbol occurs at most once in a set,
every symbol occurs in exactly
the
n
v
x
r
sets, and
symbols will be arranged symmetrically in a
m square array with blank diagonals as follows:
x
1
2
n-l
1
x
n
2n-3
2
n
x
3n-6
2n-33n-6...
x
n-l
and two symbols occuring in the same row (or column) of
the above array occur together in
Al
sets and two symbols
not occurring in the same row (or column) occur together in
A
2
sets of the design (cf. Bose and Shtmamoto (1952),
Raghavarao (1971».
Given two sets
Sand
T, the Boolean sum of
Sand
by S + T, is the set of elements which belongs to either
T, denoted
S or
T,
b
3
but not to both.
In symbols,
8 +T
== {xix
€
8
U
T,
x
f
(1.2)
8 n T} .
In this paper we discuss the application of Boolean sums on sets
of BIB designs and
C[k,4,o,v]
configuration to get BIB and triangular
designs.
2.
BOOLEAN Sill1S OF SETS OF SYMMETRICAL BIB DESIGNS
Let
v
==
design on the
be the
v
v
b, r
= k,
A be the parameters of a symmetrical BIB
symbols of a set
sets of the designs.
E
= {1,2, •.• ,v}
e·
consider the
v(v-l)/2
Si n Sj
Si + Sj for i
~
j
for i
~ j
is 2(r-A).
is
A and
Now if we
T = Si + Sj for i < j, we can easily
ij
unsymmetrical BIB design with parameters
sets
verify that they form an
v*
Sl' S2""'Sv
8ince the design is symmetrical, it is
well known that the cardinality of sets
hence the cardinality of sets
and let
= v, b* = v(v-l)/2, r * = r(v-r), k* = 2(r-A)
,
(2.1)
As an illustration, let us consider the BIB design with parameters
v
=7
==
b
, r = 3 = k, A = 1 ,
(2.2)
whose sets are
Sl: (1, 2, 4)
8 : (2, 3, 5)
2
S3: (3, 4, 6)
54: (4, 5, 7)
S5: (5, 6, 1)
S6: ( 6, 1, 2)
S7: (7, 1, 3)
e
.
(2.3)
4
The 21 sets
of Si (i
T
(i
ij
<
= 1,2, ••• ,7)
j) obtained by taking Boolean sums of every pair
are the following
T 12 ~ (1 , 3 , 4, 5)
T
T14: (1, 2, 5, 7)
TIS: (2, 4, 5, 6)
13
: (1, 2, 3, 6)
: (1, 4, 6, 7)
T
T 23: (2, 4, 5, 6)
T
T
: (1, 2, 3, 6)
T 26 :
: (1, 2, 5, 7)
T
: (1, 3, 4, 5)
T
: (1, 4, 6, 7)
T45~
: (2, 4, 5, 6)
T
: (1, 2, 5, 7)
T
T
T
T
T
T
T
16
25
27
35
37
46
56
17
24
34
36
47
57
: (2, 3, 4, 7)
: (2, 3, 4, 7)
(3, 5, 6, 7)
: (3, 5, 6, 7)
: (2, 3, 4, 7)
(1, 4, 6, 7)
: (1, 3, 4, 5)
: (3, 5, 6, 7)
(2.4)
which form a BIB design with parameters
v*
= 7, b* = 21, r *
- 12, k*
= 4, A* = 6
•
(2.5)
However, one can notice that (2.4) is obtained by repeating the sets
T12 , T13 , T14 , TIS' T16 , T17 and T and without repetitions, the sets
26
T12 , T , T , TIS' T , T and T
13
14
16
17
26
v* = 7
= b,*
r *= 4
= k,*
form a BIB design with parameters
A*= 2 .
(2.6)
The BIB design with parameters
v*
= 13, b* = 78, k* = 6, r * = 36, A* = 15
,
(2.7)
can be obtained as Boolean sum of sets of BIB design with parameters
v
= 13 = b,
r
= 4 = k,
A=1 ,
(2.8)
5
and can also be obtained by duplicating the sets of the BIB design with
parameters
= 13,
Vi
two more times.
b'
= 26,
r'
= 12,
k'
= 6,
=5
A'
(Z.9)
,
However, both the solutions will be non-isomorphic,
in the sense that we cannot obtain one solution from another by renumbering the symbols and sets, as there are no repeated sets in the
solution obtained as Boolean sum of sets of design with parameters (Z.8).
3.
TRIAnGULAR DESIGNS FROH IRREDUCIBLE BIn DESIGNS
Let
(~)
=
v, b
V-I)
= (k-l
r
(3.1)
'
be parameters of an irreducible BIB design on v
E: {l,Z, •.• ,v}
and let
the set of indices of
(i j)
Sl' 8 2 "" ,8 b
S
~
be its
symbols of a set
b sets.
Let
T
i
be
in which the i-th symbol occurs.
We now form
v*
(i < j, i, j
= 1,Z, ••. ,v).
= v(v-1)/2
symbols by taking pairs of the form
We form
b
new sets
Si, Si "",Sb
where the symbol (i,j) occurs in the sets 8"s whose indices are T + T .
i
j
The sets
Si, S2' •••
'S~
thus formed can be verified to be a triangular
design with parameters
v*
= v(v-l)!Z,
b*
= b,
r*
= Z(v-Z),
k*
* = (V-Z)
k-l'
AZ
Al
= k(v-k)
* = 4(v-4)
k-Z
(3.Z)
where the v(v-I)/2 symbols are arranged according to the array
e
x
lZ
13
Iv
lZ
x
23
Zv
13
23
x
3v
Iv
Zv
3v
x
(3.3)
6
As an illustration, let us consider the irreducible BIB design
with v = 5, k = 3 whose sets are:
8 : (1, 2, 3)
8 : (1, 2, 4)
8 3 = (1, 2, 5)
8 4 = (1, 3, 4)
8 : (1, 3, 5)
8 : (1, 4, 5)
6
8 : (2, 3, 4)
8 : (2, 3, 5)
8 : (2, 4, 5)
9
1
2
5
7
8
8
10
: (3, 4, 5)
.
(3.4)
From the method discussed in this section, we produce the following
10 sets:
e·
8 ' . (14, 15, 24, 25, 34, 35)
I'
8' , (13, 15, 23, 25, 34, 45)
2'
8 I . (13, 14, 23, 24, 35, 45)
3'
8 I , (12, 15, 23, 35, 24, 45)
4'
S' , (12, 14, 23, 34, 25, 45)
5'
8I
8 I , (12, 25, 13, 35, 14, 45)
7'
8' • (12, 24, 13, 34, 15, 45)
8'
8' • (12, 23, 14, 34, 15, 35)
9'
8' . (13, 23, 14, 24, 15, 25)
•
6' (12, 13, 24, 34, 25, 35)
10'
(3,5)
which can be seen to be a triangular design with parameters
v
* = 10 = b*
r*
, A*
2
5 x 5 square
= 6 = k*
Al*
when the symbols are arranged in a
x
12
13
14
15
12
x
23
24
25
13
x
34
14
23
24
34
x
35
45
15
25
35
45
x
=3
=4
(3,6)
;
(3.7)
The design wi th parameters (3.6) can also be constructed by forming 10
blocks
for
i < j
and i, j
= 1,2, ••• ,5
where
8*
ij
contains
7
e
the other symbols occurring in the rows of (3.7) where
ij
symbol occurs
as follows:
8* : (13, 14, 15, 23, 24, 25)
12
8* : (12, 14, 15, 23, 34, 35)
13
8* : (12, 13, 14, 25, 35, 45)
15
8* : (12, 23, 25, 13, 34, 45)
24
14, 24, 45)
8*
34 : (13, 23, 35,
8*
14 : (12, 13, 15, 24, 34, 45)
8* : (12, 24, 25, 13, 34, 35)
23
8* : (12, 23, 24, 15, 35, 45)
25
8* : (13, 23, 34, 15, 25, 45)
35
*45 :
(14, 24, 34, 15, 25, 35)
8
(3.8)
The solution (3.8) is the one given in the Tables (1954) and the solution
obtained by our present method is non-isomorphic to the known solution.
4.
e-
Let
Let
TRIANGULAR DESIGN8 FROM C[k, 4, 0; v] CONFIGURATIONS
8 , 8 ,
1
2
... , 8b
x, y, z and u
sets of a
C[k, 4, 0, v] configuration.
be any 4 symbols and let
~lOOO' ~OlOO'~OOlOand'~OOOl
be the
be the number of sets in which
y
occurs in the absence of
x, y, u ; and
u
~llOl' ~lOll' ~Olll'
x, z, u ;
and
symbols occur together
~llil.
(i
A3
~llil
y, z, u ;
z occurs in the absence of
x, y, z.
Analogously, we
~llOO' ~lOlO' ~lOOl' ~OllO' ~OlOl' ~OOll' ~lllO'
each symbol occurs and let
+
x occurs in the absence of
occurs in the absence of
introduce the notation
~lllO
b
Ai
Let
r
be the number of sets in which
be the number of sets in which any
= 2,3).
Then we have the following relations:
= 6(v-3)/(k-3)
= ~Olll +
~llil
i
= ~lOll +
(4.1)
~1111
= ~llOl +
~1111
= A3
8
+ PllOl + PlllO + PIll 1 = Xz
"1100
PlOOO
'
etc.
+ PlOOl + PlOlO + PlOll + PllOO + PllOl + PlllO + Plill
=r
By following the method of Section 3, we can generate
SI
i
, etc.
b(b-l) sets
which can be verified to form a triangular design with parameters
v*
= v(v-l)/Z, b* = b,
r*
= Z(r-X Z)' k* = (v-k)k,
*
Xl
= PlOOO
X*z
= PlOlO +
+ PlOOl + POlIO + POlll
PlOOl
+ POlIO +
~0101
when the Symbols are arranged in an array given by (3.3).
e-
5.
GENERALIZATIONS
The concept of Boolean sum of two sets
Sand
T, can be generalized
to define Boolean sum of three sets R, Sand T, denoted by R + S + T,
to consist of all elements belonging to exactly one of the three sets or
to all three sets.
R+ S+ T
In symbols,
= {xix
€
R, x
f
S,T} u {xix
€
s, x ¢ R,T}
u {xIx
€
T, x
i R,S} u {xix
€
R n S n T }
Then the method of Section 4 can be generalized and from
configuration we can generate
(a < f3 < y a,f3,y
= 1,Z, ... ,v)
such a way that the symbol
scripts belong to
v(v-l)(v-2)/6 symbols
and form
af3y
b
sets
T
a
C[k,5,o,v]
af3y
* SZ,,,.,Sb
*
* in
51'
occur in the sets
Ta + Tf3 + Ty , where
(5.1)
*.
S 's whose sub-
is the set of indices of
9
sets
SIS in which the symbol
a
occurs.
Such designs can be seen to
be extended triangular designs as defined by John (1966).
The Boolean sums of sets of partially balanced incomplete block
(PBIB) designs of Bose and Nair (1939), which are linked block designs
as defined by Roy and Laha (1956) can again be verified to be PBIB
designs.
One can take the Boolean sums of sets of PBIB designs with certain
subsets of the
v
symbols to get new PBIB designs.
For example, the
Boolean sum of each of the groups of association scheme with each of the
sets of a semi-regular group divisible design (cf. Bose and Connor (1952),
Raghavarao (1971»
will result in a group divisible design.
REFERENCES
[1]
Bose, R.C. and Connor, \'/.S. (1952), llConbinatorial properties of
group divisible incomplete block designs, Ii Ann. Math. Statist."
'23, 367-383.
[2]
Bose, R.C. and Nair, K.R. (1939), "Partially balanced incomplete
block designs," Sankhyu, 4, 337-372.
[3]
Bose, R.C. and Shimamoto, T. (1952), "Classification and analysis of
partially balanced incomplete block designs,Zl J. Amer. Statist.
Assoo." 47, 151-184.
[4]
Bose, R.C., Clatworthy, H.R. and Shrikhande, S.S. (1952), "Tables
of partially balanced designs with two associate classes," Institute
of Statistics of the University of North Carolina, Mimeo Series
No. 50.
[5]
John, P.W.M. (1966), HAn extension of the triangular association scheme
to three associate classes," J. Roy. Statist. Soo." B28, 361-365.
[6]
Raghavarao, D. (1971)" Construotions and CombinatoriaZ ProbZems in
Design of Experiments, John Wiley & Sons.
[7]
Roy, J. and Laha, R.G. (1956), "Classification and analysis of
linked block designs," Sankhy-a:" 17, 115-132.