Boolean Algebra - Etsu

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$Practice for Final Exam - MATH 1720 Precalculus -- Fall 2003 - Etsu$

90 Computer Organization and Design Fundamentalsrepresenting the AND, OR, and NOT gates. These boolean expressionscan be used to describe or evaluate the output of a circuit.There is an additional benefit. Just like algebra, a set of rules existthat when applied to boolean expressions can dramatically simplifythem. A simpler expression that produces the same output can berealized with fewer logic gates. A lower gate count results in cheapercircuitry, smaller circuit boards, and lower power consumption.If your software uses binary logic, the logic can be represented withboolean expressions. Applying the rules of simplification will make thesoftware run faster or allow it to use less memory.The next section describes the representation of the three primarylogic functions, NOT, AND, and OR, and how to convertcombinational logic to a boolean expression.5.2 Symbols of Boolean AlgebraAnalogous behavior can be shown between boolean algebra andmathematical algebra, and as a result, similar symbols and syntax canbe used. For example, the following expressions hold true in math.0 · 0 = 0 0 · 1 = 0 1 · 0 = 0 1 · 1 = 1This looks like the AND function allowing an analogy to be drawnbetween the mathematical multiply and the boolean AND functions.Therefore, in boolean algebra, A AND'ed with B is written A · B.AX = A · BBFigure 5-2 Boolean Expression for the AND FunctionMathematical addition has a similar parallel in boolean algebra,although it is not quite as flawless. The following four mathematicalexpressions hold true for addition.0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 2The first three operations match the OR function, and if the lastoperation is viewed as having a non-zero result instead of the decimalresult of two, it too can be viewed as operating similar to the OR
Chapter 5: Boolean Algebra 91function. Therefore, the boolean OR function is analogous to themathematical function of addition.ABX = A + BFigure 5-3 Boolean Expression for the OR FunctionAn analogy cannot be made between the boolean NOT and anymathematical operation. Later in this chapter we will see how the NOTfunction, unlike AND and OR, requires its own special theorems foralgebraic manipulation. The NOT is represented with a bar across theinverted element.AX = AFigure 5-4 Boolean Expression for the NOT FunctionThe NOT operation may be used to invert the result of a largerexpression. For example, the NAND function which places an inverterat the output of an AND gate is written as:X = A · BSince the bar goes across A · B, the NOT is performed after the AND.Let's begin with some simple examples. Can you determine theoutput of the boolean expression 1 + 0 + 1? Since the plus-signrepresents the OR circuit, the expression represents 1 or 0 or 1.1011Figure 5-5 Circuit Representation of the Boolean Expression 1+0+1Since an OR-gate outputs a 1 if any of its inputs equal 1, then1 + 0 + 1 = 1.The two-input XOR operation is represented using the symbol ⊕,but it can also be represented using a boolean expression. Basically, the
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Boolean Algebra - Etsu

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