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This is a little like middle-school algebra, isn t it The rules of algebra can, with certain adjustments, be directly applied to digital logic operations. This is called Boolean algebra, named after the mathematician Boole who first devised it as an alternative to truth tables.
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564 Basic digital principles
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Using Boolean representations for logic operations, some of the mathematical properties of multiplication, addition, and negation can be applied to form Boolean equations. The logical combinations on either side of any equation are equivalent. In some ways, Boolean algebra differs from conventional algebra. You must use logic rules rather than regular rules for addition, additive inverse (negation), and multiplication. Using these rules, certain facts, called theorems, can be derived. Boolean theorems all take the form of equations. Some common Boolean theorems are listed in Table 30-7.
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Table 30-7 Common theorems in Boolean algebra.
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Z) Associativity of OR Associativity of AND Distributivity DeMorgan s Theorem DeMorgan s Theorem
Boolean algebra is less messy than truth tables for designing and evaluating logic circuits. Some engineers prefer truth tables because the various logic operations are easier to envision, and all the values are shown for all logic states in all parts of a digital circuit. Other engineers would rather not deal with all those ls and 0s, nor cover whole tabletops with gigantic printouts. Boolean algebra gets around that. For extremely complex logical circuits, computers are used as an aid in design. They re good at combinatorial derivations and optimization problems that would be uneconomical (besides tedious) if done by a salaried engineer.
The flip-flop
So far, all the logic gates discussed have outputs that depend only on the inputs. They are sometimes called combinational logic gates, because the output state is simply a function of the combination of input states.
The flip-f;op 565 A flip-flop is a form of sequential logic gate. In a sequential gate, the output state depends on both the inputs and the outputs. The term sequential comes from the fact that the output depends not only on the current states, but on the states immediately preceding. A flip-flop has two states, called set and reset. Usually, the set state is called logic 1, and the reset state is called logic 0. There are several different kinds of flip-flop.
In schematic diagrams, a flip-flop is usually shown as a rectangle with two or more inputs and two outputs. If the rectangle symbol is used, the letters FF, for flip-flop, are printed or written inside at the top. The inputs of an R-S flip-flop are labeled R (reset) and S (set). The outputs are Q and Q. (Often, rather than Q, you will see Q , or perhaps Q with a line over it.) As their symbols imply, the two outputs are always in logically opposite states. The symbol for an R-S flip-flop is shown in Fig. 30-5.
30-5 Schematic symbol for an R-S flip-flop.
In an R-S flip-flop, if R 0 and S 0, the output states do not change; they stay at whatever values they re already at. If R 0 and S 1, then Q 1 and Q 0. If R 1 and S 0, then Q 0 and Q 1. That is, the Q and Q outputs will attain these values, no matter what states they were at before. But if S 1 and R 1, things get bizarre. The flip-flop becomes unpredictable. Because of this, engineers avoid letting logic ls get into both inputs of an R-S flip flop. You want logic, not absurdity! Table 30-8 is the truth table for an R-S flip-flop.
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