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2 Combinational Circuit Design

2.2 Minterms and Maxterms

There are some special Boolean Functions that - as can be seen just looking at the corresponding K-Maps - are built up in a very simple way. These are those functions that have only one single value '0' or '1'. By changing only this one function value these functions are transformed into the so-called 0-Function or 1-Function.

Therefore the function that contains only one '1' can be considered a minimal function, the one with only one '0' correspondingly a maximum function.

These special functions can be formed for an arbitrary number of arguments.


These functions are normal Boolean functions, but sometimes this fact is overlooked. In this case these functions are just referred to as Minterm and Maxterm, respectively.

In a K-Map these Boolean functions can be presented as follows:

As Minterm and Maxterm functions are determined by a single field, it would be sufficient for their characterization to indicate the type of term and the index of the distinguishing field. Common is the following abbreviated notation:

The index i marks the field under consideration. It is determined by the values of the arguments (x,...,c,b,a).

Indexing Minterms and Maxterms:

Figure 2.15: Indexing the Minterm and Maxterm Functions.


Minterms and Maxterms are characterized by single fields and can therefore be described algebraically using:


Figure 2.16: Production of Minterm and Maxterm Functions.

Using the abbreviations introduced above, the two functions in this example can be described algebraically as follows:


The example function can now be described with the aid of Minterms and Maxterms, respectively:

Karnaugh Diagram of the result function (see above):

Figure 2.17: Example Function (Maxterm/Minterm description).

As can be seen from this K-Map, the Boolean function can be described using seven (7) Minterms or one (1) Maxterm.

7 Minterme:
1 Maxterm:

or, in short form:


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