1、Chapter 2ElectronicsSection 2 Boolean Algebra for Digital Systemsn Textn New Words and Expressionsn Exercisesn Endn Selection of Word Meaning Section 2 Boolean Algebra for Digital SystemsIntroduction The mathematics of computers and other digital electronic devices have been developed from the decis
2、ive work of George Boole (l815 l864) and many others, who expanded and improved on his work. The body of thought that is known collectively as symbolic logic established the principles for deriving mathematical proofs and singularly modified our understanding and the scope of mathematics. Section 2
3、Boolean Algebra for Digital SystemsOnly a portion of this powerful system is required for our use. Boole and others were interested in developing a systematic means of deciding whether a proposition in logic or mathematics was true or false, but we shall be concerned only with the validity of the ou
4、tput of digital devices. True and false can be equated with one and zero, high and low, or on and off. These are the only two states of electrical voltage from a digital element. Thus, in this remarkable algebra performed by logic gates, there are only two values, one and zero; anySection 2 Boolean
5、Algebra for Digital Systemsalgebraic combination or manipulation can yield only these two values. Zero and one are the only symbols in binary arithmetic. The various logic gates and their interconnections can be made to perform all the essential functions required for computing and decision-making.
6、In developing digital systems the easiest procedure is to put together conceptually the gates and connections to perform the assigned task in the most direct way. Boolean algebra is then used to reduce the complexity of the system, if possible, Section 2 Boolean Algebra for Digital Systemswhile reta
7、ining the same function. The equivalent simplified combination of gates will probably be much less expensive and less difficult to assemble.Rules of Boolean algebra for digital devicesBoolean algebra has three rules of combination, as any algebra must have: the associative, the commutative, and the
8、distributive rules. To show the features of the algebra we use the variables A, B, C, and so on. To write relations between variables each one of which may take the value 0 or l, we use to mean “not A,” so if A = l , then = 0. TheSection 2 Boolean Algebra for Digital Systemscomplement of every varia
9、ble is expressed by placing a bar over the variable; the complement of= “not B“. Two fixed quantities also exist. The first is identity, I = l; the other is null, null = 0. Boolean algebra applies to the arithmetic of three basic types of gates: an OR-gate, an AND-gate and the inverter. The symbol a
10、nd the truth tables for the logic gates are shown in Fig.2-3, the truth table illustrate that the AND-gate corresponds to multiplication, the OR-gate corresponds to addition, and the inverter yield the complement of its input variable. Section 2 Boolean Algebra for Digital SystemsFig.2-3 Logic symbo
11、ls and truth tables for AND, OR, NOT(a) AND； (b) OR； (c) NOTSection 2 Boolean Algebra for Digital SystemsWe have already found thatAB = “A AND B“for the AND-gate andA + B = “A OR B“for the OR-gate.The AND, or conjunctive, algebraic form and the OR, or disjunctive, algebraic form must each obey the t
12、hree rules of algebraic combination. In the equations that follow, the reader may use the two possible values 0 and l for the variables A, B, and CSection 2 Boolean Algebra for Digital Systemsto verify the correctness of each expression. Use A = 0, B = 0, C = 0; A = l, B = 0, C = 0; and so on, in ea
13、ch expression. The associative rules state how variables may be grouped.For AND (AB)C = A(BC) = (AC)B,and for OR (A + B) + C = A + (B + C) = (A + C) + Bthe rules indicate that different groupings of variables may be used without altering the validity of the algebraic expression. The commutative rules state the order of variables.For AND AB = BA
