4.3.1 Definition of Minsets
Let B1 and B2 be subsets of a set A. Notice that the Venn diagram of Fig-
ure 4.3.1is naturally partitioned into the subsetsA1,A2,A3, andA4. Further
we observe thatA1,A2, A3, andA4 can be described in terms of B1 andB2
as follows:
Figure 4.3.1: Venn Diagram of Minsets
A1=B1∩B2c
A2=B1∩B2
A3=B1c∩B2
A4=B1c∩Bc2
Table 4.3.2: Minsets generated by two sets
Each Ai is called a minset generated by B1 and B2. We note that each
minset is formed by taking the intersection of two sets where each may be either Bk or its complement, Bkc. Note also, given two sets, there are 22 = 4
minsets.
Minsets are occasionally calledminterms.
The reader should note that if we apply all possible combinations of the operations intersection, union, and complementation to the setsB1 andB2 of
CHAPTER 4. MORE ON SETS 86 Figure 1, the smallest sets generated will be exactly the minsets, the minimum sets. Hence the derivation of the term minset.
Next, consider the Venn diagram containing three sets, B1, B2, and B3.
Draw it right now and count the regions! What are the minsets generated by B1, B2, and B3? How many are there? Following the procedures outlined
above, we note that the following are three of the 23 = 8 minsets. What are the others?
B1∩B2∩B3c
B1∩B2c∩B3
B1∩B2c∩B3c
Table 4.3.3: Three of the minsets generated byB1,B2, andB3
Definition 4.3.4 Minset. Let{B1, B2, . . . , Bn}be a set of subsets of setA.
Sets of the formD1∩D2∩ · · · ∩Dn, where eachDi may be eitherBiorBci, is
called a minset generated byB1, B2,... andBn. ♢
Example 4.3.5 A concrete example of some minsets. Consider the following concrete example. LetA={1,2,3,4,5,6}with subsetsB1={1,3,5}
andB2 ={1,2,3}. How can we use set operations applied toB1 andB2 and
produce a list of sets that contain elements ofAefficiently without duplication? As a first attempt, we might try these three sets:
B1∩B2={1,3}
B1c={2,4,6}
Bc
2={4,5,6}.
Table 4.3.6
We have produced all elements of Abut we have 4 and 6 repeated in two sets. In place ofBc
1 andB2c, let’s tryB1c∩B2 andB1∩B2c, respectively:
Bc
1∩B2={2}and
B1∩B2c={5}.
Table 4.3.7
We have now produced the elements 1, 2, 3, and 5 usingB1∩B2,B1c∩B2
andB1∩B2c yet we have not listed the elements 4 and 6. Most ways that we
could combineB1andB2such asB1∪B2orB1∪Bc2will produce duplications
of listed elements and will not produce both 4 and 6. However we note that Bc
1∩Bc2 ={4,6}, exactly the elements we need. Each element ofA appears
exactly once in one of the four minsetsB1∩B2,B1c∩B2,B1∩B2candB1c∩Bc2.
Hence, we have a partition ofA. □
4.3.2 Properties of Minsets
Theorem 4.3.8 Minset Partition Theorem. Let A be a set and let B1,
B2. . . ,Bn be subsets ofA. The set of nonempty minsets generated byB1,B2
. . .,Bn is a partition ofA.
CHAPTER 4. MORE ON SETS 87 One of the most significant facts about minsets is that any subset ofAthat can be obtained fromB1,B2. . .,Bn, using the standard set operations can be
obtained in a standard form by taking the union of selected minsets.
Definition 4.3.9 Minset Normal Form. A set is said to be in minset normal form when it is expressed as the union of zero or more distinct nonempty
minsets. ♢
Notes:
• The union of zero sets is the empty set,∅.
• Minset normal form is also calledcanonical form.
Example 4.3.10 Another Concrete Example of Minsets. Let U =
{−2,−1,0,1,2},B1={0,1,2}, and B2={0,2}. Then B1∩B2={0,2} Bc 1∩B2=∅ B1∩B2c={1} Bc 1∩B2c={−2,−1} Table 4.3.11
In this case, there are only three nonempty minsets, producing the partition
{{0,2},{1},{−2,−1}}. An example of a set that could not be produced from justB1 and B2 is the set of even elements of U, {−2,0,2}. This is because
−2 and−1cannot be separated. They are in the same minset and any union of minsets either includes or excludes them both. In general, there are23= 8
different minset normal forms because there are three nonempty minsets. This means that only 8 of the25 = 32 subsets of U could be generated from any
two setsB1 andB2. □
4.3.3 Exercises for Section 4.3
1. Consider the subsetsA ={1,7,8}, B ={1,6,9,10}, and C={1,9,10}, whereU ={1,2, ...,10}.
(a) List the nonempty minsets generated by A, B, andC.
(b) How many elements of the power set of U can be generated byA, B, and C? Compare this number with | P(U)|. Give an example of one subset that cannot be generated byA,B, and C.
2.
(a) Partition {1,2, ....9} into the minsets generated by B1 = {5,6,7},
B2={2,4,5,9}, andB3={3,4,5,6,8,9}.
(b) How many different subsets of {1,2, ...,9} can you create using B1, B2, andB3 with the standard set operations?
(c) Do there exist subsetsC1, C2, C3whose minsets will generate every
subset of{1,2, ...,9}?
3. Partition the set of strings of 0’s and 1’s of length two or less, using the minsets generated by B1 = {s | shas length2}, and B2 = {s |
CHAPTER 4. MORE ON SETS 88 4. LetB1, B2, andB3be subsets of a universal setU,
(a) Symbolically list all minsets generated by B1, B2, andB3.
(b) Illustrate with a Venn diagram all minsets obtained in part (a). (c) Express the following sets in minset normal form: B1c, B1 ∩B2 ,
B1∪B2c.
5.
(a) Partition A= {0,1,2,3,4,5} with the minsets generated by B1 =
{0,2,4} andB2={1,5}.
(b) How many different subsets ofAcan you generate fromB1andB2?
6. If{B1, B2, . . . , Bn} is a partition of A, how many minsets are generated
byB1, B2, . . . , Bn?
7. ProveTheorem 4.3.8
8. LetSbe a finite set ofnelements. LetBi„ i = 1, 2, \ldots ,kbe nonempty
subsets of S. There are 22k minset normal forms generated by the k
subsets. The number of subsets ofS is2n. Since we can make22k >2n
by choosing k ≥log2n, it is clear that two distinct minset normal-form expressions do not always equal distinct subsets ofS. Even fork <log2n, it may happen that two distinct minset normal-form expressions equal the same subset of S. Determine necessary and sufficient conditions for distinct normal-form expressions to equal distinct subsets ofS.
