274 DIGITAL BUILDING BLOCKS AND COMPUTER SYSTEMS
TABLE 6.1.1Representation of Numbers in Different Systems
Base 10 Base 2 Base 8 Base 16
(Decimal) (Binary) (Octal) (Hexadecimal) BCD00 00000 00 00 0000 0000
01 00001 01 01 0000 0001
02 00010 02 02 0000 0010
03 00011 03 03 0000 0011
04 00100 04 04 0000 0100
05 00101 05 05 0000 0101
06 00110 06 06 0000 0110
07 00111 07 07 0000 0111
08 01000 10 08 0000 1000
09 01001 11 09 0000 1001
10 01010 12 0A 0001 0000
11 01011 13 0B 0001 0001
12 01100 14 0C 0001 0010
13 01101 15 0D 0001 0011
14 01110 16 0E 0001 0100
15 01111 17 0F 0001 0101
16 10000 20 10 0001 0110
17 10001 21 11 0001 0111
18 10010 22 12 0001 1000
19 10011 23 13 0001 1001blocks:logic (combinational) blocksandsequential blocks(known asflip-flops). Logic blocks
are electronic circuits that usually have several inputs and one output. The physical devices that
perform the basic Boolean operations are known as logic gates. The basic logic gates are OR,
AND, and NOT, the interconnections of which form a logic network (also known as combinational
network), which does not generally contain memory devices. The Boolean function describing a
combinational network can be easily derived by systematically progressing from the input(s) to
the output on the logic gates.
The logic OR gate is an electronic circuit realization of the Boolean OR operation, which is
represented by the symbol+.F=A+B, for example, is read as “Fis equal toAORB.” The
OR operation yields 1 if the value ofanyof its arguments is 1. Figure 6.1.1 shows a two-input OR
gate along with itstruth table(table of combinations for the gate) and an illustration of the OR
operation. A three-input OR gate withF=A+B+Cwould have a truth table with eight entries,
which can be developed by the reader as an exercise.
The logic AND gate is an electronic circuit realization of the Boolean AND operation, which
is represented by the symbol·or by the absence of an operator. Thus, for example,F=A·B
(also written simply asF=AB) is read as “Fis equal toAANDB.” The AND operation yields 1
if and only if the values of all its arguments are 1s. Figure 6.1.2 shows a two-input AND gate, its
truth table, and an illustration of the AND operation. A three-input AND gate withF=A·B·C
would have a truth table with 2^3 =8 possible entries, which can be developed by the reader as a
desirable exercise.
The logic NOT gate is an electronic circuit realization of the Boolean NOT operation, which
is represented by a bar over the variable.F =A ̄, for example, is read as “Fequals NOTA.”
This operation is also known as thecomplementoperation. Because the NOT operation is a unary
operation and simply inverts a switching variable, the NOT gate has only one input and one
output, and is also known as aninverter. Figure 6.1.3 shows a NOT gate and its truth table.
The AND operation followed by the NOT operation inverts the output coming out of the
AND gate, and is known as NAND gate, which is typically shown in Figure 6.1.4 along with its