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Basic logic operations
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Perhaps you ve taken sentenial logic, also called propositional logic or propositional calculus (it sounds a lot more sophisticated than it really is) in high school or college. Electronic digital logic is basically the same thing. In logic, there are three basic operations. These are depicted in Table 30-2.
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Table 30-2.
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Operation NOT AND OR
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Logic operations.
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Symbology X X XY X Y X Y
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Description Negation or complement of X Conjunction of X, Y Disjunction of X, Y
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The logic NOT, also called negation or inversion, involves only a single logic variable. It changes the value of a bit from 1 to 0, or from 0 to 1, as shown in the truth table for logic operations (Table 30-3). It is represented by a minus sign in front of a capital letter, usually from the end of the alphabet (W, X, Y, or Z). It might also be represented by a squiggly dash before the letter, an apostrophe after the letter, or a line over the letter.
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Table 30-3. Truth table for logic operations.
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X 0 0 1 1 Y 0 1 0 1 -X 1 1 0 0 XY 0 0 0 1 X+Y 0 1 1 1
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The logic AND, or conjunction, involves two variables. The resultant is high if, but only if, both variables are high. If either variable is low, or if both are low, then the resultant is
560 Basic digital principles low. This operation gets its name from the fact that it corresponds to the function of the word and in propositional calculus. It is usually represented as multiplication. See Table 30-3.
The logic OR, or disjunction, also involves two variables. The resultant is high if either of the variables are high. The resultant is low if, but only if, both variables are low. This is sometimes called the inclusive OR operation. It corresponds to the word or in propositional calculus. It is represented as addition. See Table 30-3.
Table 30-4.
X 0 0 1 1 Y 0 1 0 1 XOR 0 1 1 0
AM FL Y
More logic operations.
NAND 1 1 1 0 NOR 1 0 0 0
Occasionally you ll hear about the exclusive OR or XOR operation. The resultant of X XOR Y is high if X and Y have opposite states. The resultant X XOR Y is low if X and Y have the same state. It can be thought of as either/or, but not both. The X means exclusive. The XOR operation is shown in Table 30-4.
XNOR 1 0 0 1
NAND
The logic NAND is a combination of two operations, NOT and AND. It involves two variables. The AND is performed first, and the result is negated. The value of X NAND Y is low if, but only if, both X and Y are high. Otherwise, X NAND Y is high. This is shown in Table 30-4.
The logic NOR combines NOT and OR. It involves two variables. The OR is done first, and then the result is negated. The value of X NOR Y is high if, but only if, X and Y are low. Otherwise, X NOR Y is low (Table 30-4).
XNOR
The logic XNOR combines NOT and XOR. As with all the basic logic operations except NOT, it involves two variables. The XOR is done first, and then the result is negated. This is shown in Table 30-4.
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Complex logic operations 561
Symbols for logic gates
A logic gate is an electronic switch that performs a logic operation. The earliest logic gates were built using vacuum tubes; later, transistors and diodes were employed. Modern logic gates are fabricated on integrated circuits (ICs), with hundreds or even thousands of gates per chip. An inverter (NOT gate) has one input and one output. Other logic gates usually have two inputs and a single output. The symbols for the various logic gates are shown in Fig. 30-3.
30-3 Symbols for logic gates: NOT (A), AND ( ), OR (C), XOR (D), NAND (E), NOR (F), and XNOR (G).
It is possible for AND, NAND, OR, or NOR gates to have three or more inputs. For an AND gate, the output is high if and only if all of the inputs are high. For an OR gate, the output is low if and only if all the inputs are low. The multiple-input NAND and NOR gates perform the conjunction or disjunction operation first, followed by negation. Figure 30-4 shows schematic symbols for four-input AND four-input NOR gates. The XOR and XNOR operations are hard to define when there are three or more inputs. These fall into the category of complex logic operation.
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