Minterms  (Page 2/4)

Suppose, for example, the answers are: Yes, No, No, Yes. Then ω is in the minterm $A{B}^c{C}^{c}D$ . In a similar way, we can determine the membership of each ω in the basic space. Thus, the minterms form a partition . That is, the minterms represent mutually exclusive events, one of which is sure to occur on each trial. The membership of any mintermdepends upon the membership of each generating set $A,B,C$ or D , and the relationships between them. For some classes, one or more of the minterms are empty(impossible events). As we see below, this causes no problems.

An examination of the development above shows that if we begin with a class of n events, there are 2 ⁿ minterms. To aid in systematic handling, we introduce a simple numbering system for the minterms, which we illustrateby considering again the four events $A,B,C,D$ , in that order. The answers to the four questions above can be represented numerically by the scheme

No $\sim 0$ and Yes $\sim 1$

Thus, if ω is in $A^c{B}^c{C}^c{D}^c$ , the answers are tabulated as $0000$ . If ω is in $A{B}^c{C}^{c}D$ , then this is designated $1001$ . With this scheme, the minterm arrangement above becomes

We may view these quadruples of zeros and ones as binary representations of integers, which may also be represented by their decimal equivalents, as shownin the table. Frequently, it is useful to refer to the minterms by number. If the members of the generating class are treated in afixed order, then each minterm number arrived at in the manner above specifies a minterm uniquely. Thus, for the generating class $\{A,B,C,D\}$ , in that order, we may designate

We utilize this numbering scheme on special Venn diagrams called minterm maps . These are illustrated in [link] , for the cases of three, four, and five generating events. Since the actual content of any mintermdepends upon the sets $A,B,C$ , and D in the generating class, it is customary to refer to these sets as variables . In the three-variable case, set A is the right half of the diagram and set C is the lower half; but set B is split, so that it is the union of the second and fourth columns. Similar splits occur in the other cases.

Remark . Other useful arrangements of minterm maps are employed in the analysis of switching circuits.

Minterm maps and the minterm expansion

The significance of the minterm partition of the basic space rests in large measure on the following fact.

Minterm expansion

Each Boolean combination of the elements in a generating class may be expressed as the disjoint union of an appropriate subclassof the minterms. This representation is known as the minterm expansion for the combination.

In deriving an expression for a given Boolean combination which holds for any class $\{A,B,C,D\}$ of four events, we include all possible minterms, whether empty or not. If a minterm is empty for agiven class, its presence does not modify the set content or probability assignment for the Boolean combination.