K-Maps are a convenient way to simplify Boolean. Expressions. • They can be used for up to 4 (or 5) variables. • They are a visual representation of a truth table. SIMPLIFICATION AND MINIMIZATION OF BOOLEAN. FUNCTIONS. Introduction. The complexity of digital logic gates to implement a Boolean function is. Minimization of Boolean Functions. As discussed in the “Representation of Boolean Functions” every boolean function can be expressed as a sum of minterms or a product of maxterms. The process of simplifying the algebraic expression of a boolean function is called minimization.


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Minimizing Boolean Functions

In the K-Map at left, the entry in the top row corresponds to and the entries in the minimization of boolean function row correspond to, and respectively. The next step is to notice the physical adjacencies.

There are three such groupings, labeled A, B, and C. Examine the B grouping: We could write each of minimization of boolean function six product terms, but the easiest solution is to write the numbers as binary: The bottom row corresponds to the entries and We next consider a somewhat offbeat example not in a canonical form.

Online minimization of boolean functions

The trouble with K-Maps is that the technique is designed to be used only with expressions in canonical form. Before actually doing the K-Map, we first apply simple algebraic simplification to F. Figure 3 shows how the minterms in truth tables are placed in a Karnaugh Map grid for both 3 and 4 variable expressions.

Figure 3 Looking at the 3 variable map on the left in Fig. This arrangement means that if both minterms 0 and 4 occur in a function, the first variable the one named a in Fig. The minimization of boolean function row of the Karnaugh Map is labeled with a' and the lower row with a: Any minterms appearing in the top row have the literal a' in them, and any minterms in the bottom row have the literal minimization of boolean function in them.

At the same minimization of boolean function, note that each column has the same values for the variables b and c.

Minimisation of Boolean Functions

Also, the columns are arranged in an order so that only one variable changes value as you go from one column to the next. Thus, the first two columns differ in the value of c, the second and third columns differ in the value of b, and the third and fourth columns differ in the value of c again.

The right side of Figure 3 shows that this minimization of boolean function pattern adjacent columns differ by the value of a single variable applies to the rows of a Karnaugh map too: The first and second rows of that map differ in the value of b, the second and third differ in the value of a, the third and fourth differ in the value of b, and the first and fourth differ in the value of a.

Figure 4 minimization of boolean function our sample function drawn as a Karnaugh Map.

Minterms 1 and 2 are in the second and fourth columns of the top row, while minterms 4, 5, 7, and 6 appear from left to right in the four columns of the bottom row. Figure 4 A Karnaugh Map is used to produce a minimal sum of products implementation of an expression by drawing rectangles around groups of adjacent minterms in the map; each rectangle will minimization of boolean function to one product term, and the full expression will be constructed as the OR sum of all the product terms.

The goal is to have as few product terms as possible, which implies that each product term will account for as many minterms as possible. Here are the rules for drawing the rectangles: Every minterm must be inside at least one rectangle, but there must not be any zeros inside minimization of boolean function rectangles.

Every rectangle has to be as large as possible.


Rectangles may wrap around to include cells in both the leftmost and rightmost columns. The number of minterms enclosed in a rectangle must minimization of boolean function a power of two 1, 2, 4, 8, or 16 minterms for 4-variable maps.

Figure 5 shows the rectangles for our sample function, following the procedure outlined above:

  • Simplification Of Boolean Functions
  • Logical Adjacency
  • Minimizing Boolean Functions
  • Karnaugh Maps