• 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…

  • 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…

  • 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…

  • 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…

  • 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…

  • In designing digital circuits, the designer often begins with a truth table describing what the circuit should do. The design task is largely to determine what type of circuit will perform the function described in the truth table. While some…

  • A mathematician named DeMorgan developed a pair of important rules regarding group complementation in Boolean algebra. By group complementation, I’m referring to the complement of a group of terms, represented by a long bar over more than one variable. You…

  • What Is a XOR Gate? One element conspicuously missing from the set of Boolean operations is that of Exclusive-OR, often represented as XOR. Whereas the OR function is equivalent to Boolean addition, the AND function to Boolean multiplication, and the…

  • Let’s begin with a semiconductor gate circuit in need of simplification. The “A,” “B,” and “C” input signals are assumed to be provided from switches, sensors, or perhaps other gate circuits. Where these signals originate is of no concern in…

  • Boolean algebra finds its most practical use in the simplification of logic circuits. If we translate a logic circuit’s function into symbolic (Boolean) form, and apply certain algebraic rules to the resulting equation to reduce the number of terms and/or…

  • The Commutative Property Another type of mathematical identity, called a “property” or a “law,” describes how differing variables relate to each other in a system of numbers. One of these properties is known as the commutative property, and it applies…

  • In mathematics, an identity is a statement true for all possible values of its variable or variables. The algebraic identity of x + 0 = x tells us that anything (x) added to zero equals the original “anything,” no matter…

  • Let us begin our exploration of Boolean algebra by adding numbers together: The first three sums make perfect sense to anyone familiar with elementary addition. The last sum, though, is quite possibly responsible for more confusion than any other single…

  • Mathematical rules are based on the defining limits we place on the particular numerical quantities dealt with. When we say that 1 + 1 = 2 or 3 + 4 = 7, we are implying the use of integer quantities:…

  • Before the advent of solid-state logic circuits, logical control systems were designed and built exclusively around electromechanical relays. Relays are far from obsolete in modern design, but have been replaced in many of their former roles as logic-level control devices,…

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