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:
CHAPTER 4. MORE ON SETS 82 Table 4.3.2 Minsets generated by two sets
A1=B1∩B2c A2=B1∩B2 A3=B1c∩B2 A4=Bc
1∩Bc2
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 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?
Table 4.3.3 Three of the minsets generated by B1,B2, and B3 B1∩B2∩B3c
B1∩Bc 2∩B3 B1∩Bc
2∩B3c
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 to and produce a partition ofA? As a first attempt, we might try these three sets:
Table 4.3.6 B1∩B2={1,3} Bc 1={2,4,6} Bc 2={4,5,6}.
We have produced all elements of Abut we have 4 and 6 repeated in two sets. In place ofBc1 andB2c, let’s tryB1c∩B2 andB1∩B2c, respectively: Table 4.3.7
Bc
1∩B2={2}and B1∩B2c={5}.
We have now produced the elements 1, 2, 3, and 5 usingB1∩B2,Bc 1∩B2 andB1∩Bc
2 yet we have not listed the elements 4 and 6. Most ways that we could combineB1andB2such asB1∪B2orB1∪Bc
2will produce duplications of listed elements and will not produce both 4 and 6. However we note that B1c∩Bc2={4,6}, exactly the elements we need.
After more experimenting, we might reach a conclusion that each element ofAappears exactly once in one of the four minsetsB1∩B2,B1c∩B2,B1∩Bc2 andB1c∩Bc2. Hence, we have a partition ofA. In fact this is the finest partition
CHAPTER 4. MORE ON SETS 83 ofAin that all other partitions we could generate consist of selected unions of these minsets.
At this point, we might ask and be able to answer the question “How many different subsets of our universe can we generate fromB1andB2?” The answer is2number of nonempty minsets, which is24= 16in this case. Notice that in general, it would be impossible to find two sets from which we could generate all subsets of A = {1,2,3,4,5,6} since there will never be more than four nonempty minsets. If we allowed ourselves three subsets and tried to generat all sets from them, then the number of minsets would be23 = 8. With only six elements inA, there could be six minsets, each containing a single element. In that case we could generate the whole power set 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.
Proof. The proof of this theorem is left to the reader. ■
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 Table 4.3.11 B1∩B2={0,2} Bc 1∩B2=∅ B1∩Bc 2={1} Bc 1∩B2c={−2,−1}
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
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
CHAPTER 4. MORE ON SETS 84 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 | sstarts with a0}.
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: Bc
1, 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. LetS be a finite set ofn elements. LetBi, i= 1,2, . . . , k be 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.