Minsets

In document Applied Discrete Structures (Page 99-102)

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 and1cannot 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.

In document Applied Discrete Structures (Page 99-102)