Tag: Venn Diagrams

8.11 Larger 5 & 6variable Karnaugh Maps
Larger Karnaugh maps reduce larger logic designs. How large is large enough? That depends on the number of inputs, fanins, to the logic circuit under consideration. One of the large programmable logic companies has an answer. Altera’s own data, extracted from its library of customer designs, supports the value of heterogeneity. By examining logic cones,…

8.10 Don’t Care Cells in the Karnaugh Map
Up to this point we have considered logic reduction problems where the input conditions were completely specified. That is, a 3variable truth table or Karnaugh map had 2n = 23 or 8entries, a full table or map. It is not always necessary to fill in the complete truth table for some realworld problems. We may…

8.9 Sum and Product Notation
For reference, this section introduces the terminology used in some texts to describe the minterms and maxterms assigned to a Karnaugh map. Otherwise, there is no new material here. Σ (sigma) indicates sum and lower case “m” indicates minterms. Σm indicates sum of minterms. The following example is revisited to illustrate our point. Instead of…

8.8 Minterm vs Maxterm Solution
So far we have been finding SumOfProduct (SOP) solutions to logic reduction problems. For each of these SOP solutions, there is also a ProductOfSums solution (POS), which could be more useful, depending on the application. Before working a ProductOfSums solution, we need to introduce some new terminology. The procedure below for mapping product terms is…

8.7 Larger 4variable Karnaugh Maps
Knowing how to generate Gray code should allow us to build larger maps. Actually, all we need to do is look at the left to right sequence across the top of the 3variable map, and copy it down the left side of the 4variable map. See below. Reductions of 4 Variable K Maps The following…

8.6 Logic Simplification With Karnaugh Maps
The logic simplification examples that we have done so far could have been performed with Boolean algebra about as quickly. Real world logic simplification problems call for larger Karnaugh maps so that we may do serious work. We will work some contrived examples in this section, leaving most of the real world applications for the…

8.5 Karnaugh Maps, Truth Tables, and Boolean Expressions
Who Developed the Karnaugh Map? Maurice Karnaugh, a telecommunications engineer, developed the Karnaugh map at Bell Labs in 1953 while designing digital logic based telephone switching circuits. The Use of Karnaugh Map Now that we have developed the Karnaugh map with the aid of Venn diagrams, let’s put it to use. Karnaugh maps reduce logic…

8.4 Making a Venn Diagram Look Like a Karnaugh Map
Starting with circle A in a rectangular A’ universe in figure (a) below, we morph a Venn diagram into almost a Karnaugh map. We expand circle A at (b) and (c), conform to the rectangular A’ universe at (d), and change A to a rectangle at (e). Anything left outside of A is A’ .…

8.3 Boolean Relationships on Venn Diagrams
The fourth example has A partially overlapping B. Though, we will first look at the whole of all hatched area below, then later only the overlapping region. Let’s assign some Boolean expressions to the regions above as shown below. Below left there is a red horizontal hatched area for A. There is a blue vertical…

8.2 Venn Diagrams and Sets
Mathematicians use Venn diagrams to show the logical relationships of sets (collections of objects) to one another. Perhaps you have already seen Venn diagrams in your algebra or other mathematics studies. If you have, you may remember overlapping circles and the union and intersection of sets. We will review the overlapping circles of the Venn…

8.1 Introduction to Karnaugh Mapping
Why learn about Karnaugh maps? The Karnaugh map, like Boolean algebra, is a simplification tool applicable to digital logic. See the “Toxic waste incinerator” in the Boolean algebra chapter for an example of Boolean simplification of digital logic. The Karnaugh Map will simplify logic faster and more easily in most cases. Boolean simplification is actually…