Chapter 2 Electronics,Section 2 Boolean Algebra for Digital Systems,Text,New Words and Expressions,Exercises,End,Selection of Word Meaning,Section 2 Boolean Algebra for Digital Systems,Introduction The mathematics of computers and other digital electronic devices have been developed from the decisive work of George Boole (l815l864) 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 Boolean Algebra for Digital Systems,Only 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 output 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; any,Section 2 Boolean Algebra for Digital Systems,algebraic 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. 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 Systems,while retaining 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 devices Boolean algebra has three rules of combination, as any algebra must have: the associative, the commutative, and the 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. The,Section 2 Boolean Algebra for Digital Systems,complement of every variable 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 and 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 Systems,Fig.2-3 Logic symbols and truth tables for AND, OR, NOT (a) AND； (b) OR； (c) NOT,Section 2 Boolean Algebra for Digital Systems,We have already found that AB = “A AND B“ for the AND-gate and A + 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 three 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 C,Section 2 Boolean Algebra for Digital Systems,to verify the correctness of each expression. Use A = 0, B = 0, C = 0; A = l, B = 0, C = 0; and so on, in each 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) + B the 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,Section 2 Boolean Algebra for Digital Systems,and for OR A+B = B+A the rules indicate that the operations can be grouped and expanded as shown. Before we show the remaining rules of Boolean algebra for digital devices, let us confirm the distributive rule for AND by writing the truth table, Table 2-l. We will discover soon how we knew that we could write AB + C = (A + C)(B + C), which is proved by the truth table to be a proper expansion.,Section 2 Boolean Algebra for Digital Systems,Table 2-1 Truth table for the AND-distribution rule,Section 2 Boolean Algebra for Digital Systems,The more complex expression and its simpler form yield identical values. Because binary logic is dominated by an algebra in which a sum of ones equals one, the truth table permits us to identify the equivalence among algebraic expressions. A truth table may be used to find a simpler equivalent to a more complex relation among variables, if such an equivalent exists. We will see shortly how the reduction of com