Category: Education
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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’ .…
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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…
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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…
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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…
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7.9 Converting Truth Tables into Boolean Expressions
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 people seem to have a natural ability to look at a truth table and immediately…
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7.8 DeMorgan’s Theorems
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 should recall from the chapter on logic gates that inverting all inputs to a gate…
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7.7 The Exclusive-OR Function: The XOR Gate
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 NOT function (inverter) to Boolean complementation, there is no direct Boolean equivalent for Exclusive-OR. This…
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7.6 Circuit Simplification Examples
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 the task of gate reduction. How to Write a Boolean Expression to Simplify Circuits Our…
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7.5 Boolean Rules for Simplification
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 arithmetic operations, the simplified equation may be translated back into circuit form for a logic…
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7.4 Boolean Algebraic Properties
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 equally to addition and multiplication. In essence, the commutative property tells us we can reverse…
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7.3 Boolean Algebraic Identities
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 what value that “anything” (x) may be. Like ordinary algebra, Boolean algebra has its own…
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7.2 Boolean Arithmetic
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 statement in digital electronics, because it seems to run contrary to the basic principles of…
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7.1 Introduction to Boolean Algebra
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: the same types of numbers we all learned to count in elementary education. What most…
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6.6 Programmable Logic Controllers (PLC)
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, relegated most often to those applications demanding high current and/or high voltage switching. Systems and…
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6.5 Fail-safe Design
Logic circuits, whether comprised of electromechanical relays or solid-state gates, can be built in many different ways to perform the same functions. There is usually no one “correct” way to design a complex logic circuit, but there are usually ways that are better than others. In control systems, safety is (or at least should be)…