Karo diagram analysis of logical functions

Logical Function Karo Chart

Karnaugh map simplified logical function method

In the Karnaugh map, adjacent minimum terms are also logically adjacent. Logically adjacent means that the two minterms are the same except that one variable has a different form and is a reciprocal variable. Therefore, these adjacent minterms can be combined into an AND term and the reciprocal variables eliminated.

①Which squares are adjacent

In the Karnaugh map, there are three adjacent situations:

Connected: Two small squares are next to each other, no matter from which direction, up and down or left and right;

Relative: the small squares at both ends of any row or column;

Overlapping: small squares that overlap when folded in half.

②Principles of merging

All adjacent minimum items can be merged, so how to merge and what is the result of the merge?

(1) Combine two minimum terms, eliminate one reciprocal variable, and retain the common variable;

(2) Combine the four minimum terms, eliminate the two mutually exclusive variables, and retain the common variables;

(3) The eight minimum terms are merged, three mutually exclusive variables are eliminated, and the common variables are retained.

Generally speaking, 2^n minimum terms can be combined to eliminate n variables. When all the minimum terms in the Karnaugh map are "1", the entire Karnaugh map is a large adjacent area, which can eliminate all n reciprocal variables so that the function value is always "1".

The following principles should be followed when drawing a circle:

(1) Take the larger one and not the smaller one. The larger the circle, the more variables are eliminated and the simpler the AND term is. If you can draw it in a big circle, don’t draw it in a small circle;

(2) The fewer the circles, the fewer the simplified AND terms;

(3) A minimum term can be used repeatedly, that is, a square can be surrounded by multiple circles at the same time as long as needed;

(4) At least one small square in a circle is not surrounded by other circles;

(5) The circle must be drawn until it covers every "1" square.

Eliminate the reciprocal variables in each circle, retain the common variables, and then logically "OR" the corresponding AND terms to obtain the simplest AND-OR expression.

How to draw a Karnaugh diagram using WORD

The steps to use Karnaugh map to simplify logical functions are as follows:

Step 1: Transform the logical function into the form of the sum of minimum terms

Step 2: Draw a Karnaugh map representing the logical function

Step 3: Find the smallest term that can be merged and draw a merge circle

Step 4: Write the simplest AND-OR expression

When using Karnaugh maps to simplify logical functions, the key is to draw merge circles. The merged circles are drawn differently, and the expressions of the logical functions are also different. Therefore, you should pay attention to the following points when drawing merged circles:

①First find the isolated square 1 and draw a circle.

②The larger the range of the merge circle, the better, but it must contain (i=0,1,2,3...) 1 squares, so that more variables can be eliminated.

③The fewer the number of merging circles, the better, because the number of merging circles corresponds to the number of product terms in the simplified result. The fewer the number of circles, the fewer the AND terms in the AND-OR expression.

④Each merge circle must contain at least one square that is not included in other merge circles, so as to ensure that this merge circle is not redundant.

⑤All squares in the Karnaugh map must be circled at least once, and there must be no missing square.

In this way, by "adding" the AND terms corresponding to each merging circle, you will get the simplest AND-OR expression.

Similar method, as long as the merging circle is changed to the 0 square in the Karnaugh map, and the largest term that can be merged is found, the simplest OR-AND expression of the logical function can be obtained.

The rule of merging the largest term is basically the same as the rule of merging the smallest term. The difference is that when merging the largest items, you must find the adjacency of square 0. Each merged circle can be composed of (i=0,1,2,3...) 0 squares. Each merged circle corresponds to an OR term. The OR term is composed of the OR of variables with unchanged values ​​in the circle. Among them, the value of 0 corresponds to the original variable, and the value of 1 corresponds to the inverse variable. Then AND the corresponding OR terms of each merged circle to get the simplest OR-AND expression

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