**4.3.1 Definition of Minsets**

Let *B*1 and *B*2 be subsets of a set *A. Notice that the Venn diagram of* Fig-

ure 4.3.1is naturally partitioned into the subsets*A*1,*A*2,*A*3, and*A*4. Further

we observe that*A*1,*A*2, *A*3, and*A*4 can be described in terms of *B*1 and*B*2

as follows:

**Figure 4.3.1:** Venn Diagram of Minsets

*A*1=*B*1*∩B*2*c*

*A*2=*B*1*∩B*2

*A*3=*B*1*c∩B*2

*A*4=*B*1*c∩Bc*2

**Table 4.3.2:** Minsets generated by two sets

Each *Ai* is called a minset generated by *B*1 and *B*2. 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 called*minterms.*

The reader should note that if we apply all possible combinations of the
operations intersection, union, and complementation to the sets*B*1 and*B*2 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, *B*1, *B*2, and *B*3.

Draw it right now and count the regions! What are the minsets generated by
*B*1, *B*2, and *B*3? 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?

*B*1*∩B*2*∩B*3*c*

*B*1*∩B*2*c∩B*3

*B*1*∩B*2*c∩B*3*c*

**Table 4.3.3:** Three of the minsets generated by*B*1,*B*2, and*B*3

**Definition 4.3.4 Minset.** Let*{B*1*, B*2*, . . . , Bn}*be a set of subsets of set*A.*

Sets of the form*D*1*∩D*2*∩ · · · ∩Dn*, where each*Di* may be either*Bi*or*Bci*, is

called a minset generated by*B*1, *B*2,... and*Bn*. ♢

**Example 4.3.5** **A concrete example of some minsets.** Consider the
following concrete example. Let*A*=*{*1,2,3,4,5,6*}*with subsets*B*1=*{*1,3,5*}*

and*B*2 =*{*1,2,3*}*. How can we use set operations applied to*B*1 and*B*2 and

produce a list of sets that contain elements of*A*efficiently without duplication?
As a first attempt, we might try these three sets:

*B*1*∩B*2=*{*1,3*}*

*B*1*c*=*{*2,4,6*}*

*Bc*

2=*{*4,5,6*}*.

**Table 4.3.6**

We have produced all elements of *A*but we have 4 and 6 repeated in two
sets. In place of*Bc*

1 and*B*2*c*, let’s try*B*1*c∩B*2 and*B*1*∩B*2*c*, respectively:

*Bc*

1*∩B*2=*{*2*}*and

*B*1*∩B*2*c*=*{*5*}*.

**Table 4.3.7**

We have now produced the elements 1, 2, 3, and 5 using*B*1*∩B*2,*B*1*c∩B*2

and*B*1*∩B*2*c* yet we have not listed the elements 4 and 6. Most ways that we

could combine*B*1and*B*2such as*B*1*∪B*2or*B*1*∪Bc*2will produce duplications

of listed elements and will not produce both 4 and 6. However we note that
*Bc*

1*∩Bc*2 =*{*4,6*}*, exactly the elements we need. Each element of*A* appears

exactly once in one of the four minsets*B*1*∩B*2,*B*1*c∩B*2,*B*1*∩B*2*c*and*B*1*c∩Bc*2.

Hence, we have a partition of*A.* □

**4.3.2 Properties of Minsets**

**Theorem 4.3.8 Minset Partition Theorem.** *Let* *A* *be a set and let* *B*1*,*

*B*2*. . .* *,Bn* *be subsets ofA. The set of nonempty minsets generated byB*1*,B*2

*. . .,Bn* *is a partition ofA.*

CHAPTER 4. MORE ON SETS 87
One of the most significant facts about minsets is that any subset of*A*that
can be obtained from*B*1,*B*2*. . .,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 called**canonical form.**

**Example 4.3.10** **Another Concrete Example of Minsets.** Let *U* =

*{−*2,*−*1,0,1,2*}*,*B*1=*{*0,1,2*}*, and *B*2=*{*0,2*}*. Then
*B*1*∩B*2=*{*0,2*}*
*Bc*
1*∩B*2=*∅*
*B*1*∩B*2*c*=*{*1*}*
*Bc*
1*∩B*2*c*=*{−*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
just*B*1 and *B*2 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 sets*B*1 and*B*2. □

**4.3.3 Exercises for Section 4.3**

**1.** Consider the subsets*A* =*{*1,7,8*}*, *B* =*{*1,6,9,10*}*, and *C*=*{*1,9,10*}*,
where*U* =*{*1,2, ...,10*}*.

(a) List the nonempty minsets generated by *A, B,* and*C.*

(b) How many elements of the power set of *U* can be generated by*A,*
*B, and* *C? Compare this number with* *| P*(U)*|*. Give an example
of one subset that cannot be generated by*A,B, and* *C.*

**2.**

(a) Partition *{*1,2, ....9*}* into the minsets generated by *B*1 = *{*5,6,7*}*,

*B*2=*{*2,4,5,9*}*, and*B*3=*{*3,4,5,6,8,9*}*.

(b) How many different subsets of *{*1,2, ...,9*}* can you create using
*B*1*, B*2, and*B*3 with the standard set operations?

(c) Do there exist subsets*C*1*, C*2*, C*3whose 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 *B*1 = *{s* *|* *s*has length2*}*, and *B*2 = *{s* *|*

CHAPTER 4. MORE ON SETS 88
**4.** Let*B*1*, B*2, and*B*3be subsets of a universal set*U*,

(a) Symbolically list all minsets generated by *B*1*, B*2, and*B*3.

(b) Illustrate with a Venn diagram all minsets obtained in part (a).
(c) Express the following sets in minset normal form: *B*_{1}*c*, *B*1 *∩B*2 ,

*B*1*∪B*2*c*.

**5.**

(a) Partition *A*= *{*0,1,2,3,4,5*}* with the minsets generated by *B*1 =

*{*0,2,4*}* and*B*2=*{*1,5*}*.

(b) How many different subsets of*A*can you generate from*B*1and*B*2?

**6.** If*{B*1*, B*2*, . . . , Bn}* is a partition of *A, how many minsets are generated*

by*B*1*, B*2*, . . . , Bn*?

**7.** ProveTheorem 4.3.8

**8.** Let*S*be a finite set of*n*elements. Let*Bi*„ i = 1, 2, \ldots ,*k*be nonempty

subsets of *S. There are* 22*k* _{minset normal forms generated by the} _{k}

subsets. The number of subsets of*S* is2*n*_{. Since we can make}_{2}2*k* _{>}_{2}*n*

by choosing *k* *≥*log_{2}*n, it is clear that two distinct minset normal-form*
expressions do not always equal distinct subsets of*S. Even fork <*log_{2}*n,*
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 of*S.*