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To illustrate how the adder of Figure 6.41 works, let us see how it will add two numbers
say, 9 and 11. The binary equivalent of decimal 9 is 1001, and that of decimal 11 is 1011.
Figure 6.42 shows the binary adder with these inputs.
As shown in the figure, the half-adder adds 1 + 1 to give a sum of 0 and a carry 1. The
carry goes into the first full-adder, which adds 0 + 1 + 1 to get a sum of 0 and a carry of
1. This carry goes into the next full-adder, which adds 0 + 0 + 1 to get a sum of 1 and a
carry of 0. The last full-adder adds 1 + 1 + 0 to get a sum of 0 and a carry of 1. The final
output of the system is 10100. The decimal equivalent of binary 10100 is 20 which is the
correct decimal sum of 9 and 11.
The parallel binary adder of Figure 6.41 has limited capacity. The largest binary numbers
that can be added using it are 1111 and 1111. Hence its maximum capacity is:
In order to increase the capacity, more full-adders can be connected to the left end of the
system. For instance, to add six bit numbers, two more full-adders must be connected and
for adding eight bit numbers, four more full-adders must be connected to the left end of
the full-adder of Figure 6.41.
Points to Remember
1. Boolean algebra deals with the binary number system. That is, the variables used in
the Boolean equations may have only two possible values (0 and 1).
2. In Boolean algebra, the 'OR' operator used for logical addition is represented by the
symbol '+'; the 'AND' operator used for logical multiplication is represented by the
symbol ' • '; and the 'NOT' operator used for complementation is represented by the
3. As regards precedence of Boolean operators, 'NOT' operator takes precedence over
'AND' and 'OR' operators, and 'AND' operator takes precedence over 'OR' operator.
4. The postulates of Boolean algebra are
(a) A = 0 if and only if A is not equal to 1
(b) A = 1 if and only if A is not equal to 0
(c) x + 0 = x
(d) x • 1 = x
(e) x + y = y...
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- Spring '14