F=C'B'A + C'BA' + C'BA + CB'A' + CB'A + CBA' + CBA. The figure below shows the truth map for this function.
Each rectangle represents a term in the simplified boolean function. Therefore, the simplified boolean function will contain only three terms. You build each term using the process of elimination. You eliminate any variables whose primed and unprimed form both appear within the rectangle. Consider the long skinny rectangle above that is sitting in the row where C=1. This rectangle contains both A and B in primed and unprimed form. Therefore, we can eliminate A and B from the term. Since the rectangle sits in the C=1 region, this rectangle represents the single literal C.
Now consider the solid square above. This rectangle includes C, C', B, B' and A. Therefore, it represents the single term A. Likewise, the square with the dotted line above contains C, C', A, A' and B. Therefore, it represents the single term B.
The final, optimal, function is the sum (logical OR) of the terms represented by the three squares. Therefore, F= A + B + C. You do not have to consider squares containing zeros.
When enclosing groups of ones in the truth map, you must consider the fact that a truth map forms a torus (i.e., a doughnut shape). The right edge of the map wraps around to the left edge (and vice-versa). Likewise, the top edge wraps around to the bottom edge. This introduces additional possibilities when surrounding groups of ones in a map. Consider the boolean function F=C'B'A' + C'BA' + CB'A' + CBA'. The figure below shows the truth map for this function
At first glance, you would think that there are two possible rectangles here as the figure below shows.
However, because the truth map is a continuous object with the right side and left sides connected, we can form a single, square rectangle, as shown below:
So what? Why do we care if we have one rectangle or two in the truth map? The answer is because the larger the rectangles are, the more terms they will eliminate. The fewer rectangles that we have, the fewer terms will appear in the final boolean function. For example, the former example with two rectangles generates a function with two terms. The first rectangle (on the left) eliminates the C variable, leaving A'B' as its term. The second rectangle, on the right, also eliminates the C variable, leaving the term BA'. Therefore, this truth map would produce the equation F=A'B' + A'B. We know this is not optimal, see Th 13. Now consider the second truth map above. Here we have a single rectangle so our boolean function will only have a single term. Obviously this is more optimal than an equation with two terms. Since this rectangle includes both C and C' and also B and B', the only term left is A'. This boolean function, therefore, reduces to F=A'.
There are only two cases that the truth map method cannot handle properly: a truth map that contains all zeros or a truth map that contains all ones. These two cases correspond to the boolean functions F=0 and F=1, respectively. These functions are easy to generate by inspection of the truth map.
An important thing you must keep in mind when optimizing boolean functions using the mapping method is that you always want to pick the largest rectangles whose sides' lengths are a power of two. You must do this even for overlapping rectangles (unless one rectangle encloses another). Consider the boolean function F = C'B'A' + C'BA' + CB'A' + C'AB + CBA' + CBA. This produces the truth map appearing below:
The initial temptation is to create one of the sets of rectangles found below:
However, the correct mapping is
All three mappings will produce a boolean function with two terms. However, the first two will produce the expressions F= B + A'B' and F = AB + A'. The third form produces F = B + A'. Obviously, this last form is more optimal than the other two forms (see theorems 11 and 12).
For functions of three variables, the size of the rectangle determines the number of terms it represents:
Truth maps you create for functions of four variables are even trickier. This is because there are lots of places rectangles can hide from you along the edges. The Figures below show some possible places rectangles can hide.
This list of patterns doesn't even begin to cover all of them! For example, these diagrams show none of the 1x2 rectangles. You must exercise care when working with four variable maps to ensure you select the largest possible rectangles, especially when overlap occurs. This is particularly important with you have a rectangle next to an edge of the truth map.
As with functions of three variables, the size of the rectangle in a four variable truth map controls the number of terms it represents:
This last example demonstrates an optimization of a function containing four variables. The function is F = D'C'B'A' + D'C'B'A + D'C'BA + D'C'BA' + D'CB'A + D'CBA + DCB'A + DCBA + DC'B'A' + DC'BA', the truth map appears below:
Here are two possible sets of maximal rectangles for this function, each producing three terms
Both functions are equivalent; both are as optimal as you can get. Either will suffice for our purposes.
First, let's consider the term represented by the rectangle formed by the four corners. This rectangle contains B, B', D, and D'; so we can eliminate those terms. The remaining terms contained within these rectangles are C' and A', so this rectangle represents the term C'A'.
The second rectangle, common to both maps in the figure, is the rectangle formed by the middle four squares. This rectangle includes the terms A, B, B', C, D, and D'. Eliminating B, B', D, and D' (since both primed and unprimed terms exist), we obtain CA as the term for this rectangle.
The map on the left in the figurehas a third term represented by the top row. This term includes the variables A, A', B, B', C' and D'. Since it contains A, A', B, and B', we can eliminate these terms. This leaves the term C'D'. Therefore, the function represented by the map on the left is F=C'A' + CA + C'D'.
The map on the right in the figure has a third term represented by the top/middle four squares. This rectangle subsumes the variables A, B, B', C, C', and D'. We can eliminate B, B', C, and C' since both primed and unprimed versions appear, this leaves the term AD. Therefore, the function represented by the function on the right is F=C'A' + CA + AD'.
Since both expressions are equivalent, contain the same number of terms, and the same number of operators, either form is equivalent. Unless there is another reason for choosing one over the other, you can use either form.
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