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process is equally easy: To convert from hexadecimal to binary, replace each hexadecimal number with the equivalent four-bit binary word
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131 Convert the following decimal numbers to binary form:
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132 Convert the following binary numbers to decimal:
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1101 10111 0001101 1110111011 101100101011101 b d f h 11011 01011 0001101101 1011011001101
133 Perform the following additions and subtractions Express the answer in decimal form for problems (a) (d) and in binary form for problems (e) (h) a 100112 + 1011012 b 1001012 + 1001012 c 010112 + 011012 d 1011012 + 1001112 e 6410 3210 f 12710 6310
g 93510 427510 h
9 5 (84 32 )10 (48 16 )10
134 How many possible numbers can be represented in a 12-bit word
13
Digital Logic Circuits
135 If we use an eight-bit word with a sign bit (seven magnitude bits plus one sign bit) to represent voltages 5 V and +5 V, what is the smallest increment of voltage that can be represented 136 Convert the following numbers from hex to binary or from binary to hex:
a c e F83 A6 101110012 b d f 3C9 1101011102 110111011012 c 1011110
137 Find the two s complement of the following binary numbers:
a 11101001 b 10010111
138 Convert the following numbers from hex to binary, and nd their two s complements: a F43 b 2B9 c A6
BOOLEAN ALGEBRA
Table 139 Rules for logical addition (OR) 0+0=0 0+1=1 1+0=1 1+1=1
The mathematics associated with the binary number system (and with the more general eld of logic) is called Boolean, in honor of the English mathematician George Boole, who published a treatise in 1854 entitled An Investigation of the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities The development of a logical algebra, as Boole called it, is one of the results of his investigations The variables in a Boolean, or logic, expression can take only one of two values, usually represented by the numbers 0 and 1 These variables are sometimes referred to as true (1) and false (0) This convention is normally referred to as positive logic There is also a negative logic convention in which the roles of logic 1 and logic 0 are reversed In this book we shall employ only positive logic Analysis of logic functions, that is, functions of logical (Boolean) variables, can be carried out in terms of truth tables A truth table is a listing of all the possible values each of the Boolean variables can take, and of the corresponding value of the desired function In the following paragraphs we shall de ne the basic logic functions upon which Boolean algebra is founded, and we shall describe each in terms of a set of rules and a truth table; in addition, we shall also introduce logic gates Logic gates are physical devices (see 10) that can be used to implement logic functions AND and OR Gates The basis of Boolean algebra lies in the operations of logical addition, or the OR operation; and logical multiplication, or the AND operation Both of these nd a correspondence in simple logic gates, as we shall presently illustrate Logical addition, although represented by the symbol +, differs from conventional algebraic addition, as shown in the last rule listed in Table 139 Note that this rule also differs from the last rule of binary addition studied in the previous section Logical addition can be represented by the logic gate called an OR gate, whose symbol and whose inputs and outputs are shown in Figure 1311 The OR gate represents the following logical statement: If either X or Y is true (1), then Z is true(1) (131)
X Y OR gate X 0 0 1 1 Y 0 1 0 1 Truth table Z 0 1 1 1 Z
Figure 1311 Logical addition and the OR gate
Part II
Electronics
This rule is embodied in the electronic gates discussed in 9, in which a logic 1 corresponds, say, to a 5-V signal and a logic 0 to a 0-V signal Logical multiplication is denoted by the center dot ( ) and is de ned by the rules of Table 1310 Figure 1312 depicts the AND gate, which corresponds to this operation The AND gate corresponds to the following logical statement: If both X and Y are true (1), then Z is true (1) (132)
Table 1310 Rules for logical multiplication (AND) 0 0=0 0 1=0 1 0=0 1 1=1
One can easily envision logic gates (AND and OR) with an arbitrary number of inputs; three- and four-input gates are not uncommon The rules that de ne a logic function are often represented in tabular form by means of a truth table Truth tables for the AND and OR gates are shown in Figures 1311 and 1312 A truth table is nothing more than a tabular summary of all of the possible outputs of a logic gate, given all the possible input values If the number of inputs is 3, the number of possible combinations grows from 4 to 8, but the basic idea is unchanged Truth tables are very useful in de ning logic functions A typical logic design problem might specify requirements such as the output Z shall be logic 1 only when the condition (X = 1 AND Y = 1) OR (W = 1) occurs, and shall be logic 0 otherwise The truth table for this particular logic function is shown in Figure 1313 as an illustration The design consists, then, of determining the combination of logic gates that exactly implements the required logic function Truth tables can greatly simplify this procedure The AND and OR gates form the basis of all logic design in conjunction with the NOT gate The NOT gate is essentially an inverter (which can be constructed using bipolar or eld-effect transistors, as discussed in 10), and it provides the complement of the logic variable connected to its input The complement of a logic variable X is denoted by X The NOT gate has only one input, as shown in Figure 1314 To illustrate the use of the NOT gate, or inverter, we return to the design example of Figure 1313, where we required that the output of a logic circuit be Z = 1 only if X = 0 AND Y = 1 OR if W = 1 We recognize that except for the requirement X = 0, this problem would be identical if we stated it as follows: The output Z shall be logic 1 only when the condition (X = 1 AND Y = 1) OR (W = 1) occurs, and shall be logic 0 otherwise If we use an inverter to convert X to X, we see that the required condition becomes (X = 1 AND Y = 1) OR (W = 1) The formal solution to this elementary design exercise is illustrated in Figure 1315 In the course of the discussion of logic gates, extensive use will be made of truth tables to evaluate logic expressions A set of basic rules will facilitate this task Table 1311 lists some of the rules of Boolean algebra; each of these can be proven by using a truth table, as will be shown in examples and exercises An example proof for rule 16 is given in Figure 1316 in the form of a truth table This technique can be employed to prove any of the laws of Table 1311 From the simple truth table in Figure 1316, which was obtained step by step, we can clearly see that indeed X (X + Y ) = X This methodology for proving the validity of logical equations is called proof by perfect induction The 19 rules of Table 1311 can be used to simplify logic expressions To complete the introductory material on Boolean algebra, a few paragraphs need to be devoted to two very important theorems, called De Morgan s theorems
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