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The truth table and the block diagram symbol for the exclusiveOR and the equivalence
operations are shown in Figure 6.33 and Figure 6.34 respectively. Observe that the two
operations are the complement of each other. Each is commutative and associative.
Because of these two properties, a function of three or more variables can be expressed
without parentheses as follows:
Inputs
Output
A
B
C=A©B
0
0
0
0
1
1
1
0
1
1
1
0
(b) Truth table for ExclusiveOR.
Figure 6.33. Block diagram symbols and truth table for ExclusiveOR operation.
A
a
C = AOB
0
0
1
0
l
0
1
0
0
1
l
1
The exclusiveOR and equivalence operations have many excellent characteristics as
candidates for logic gates but are expensive to construct with physical components. They
are available as standard logic gates in IC packages but are usually constructed internally
with other standard gates. For example, Figure 6.3 5(a) shows the implementation of a
twoinput exclusiveOR function with AND, OR and NOT gates. Figure 6.35(b) shows
its implementation with NAND gates. Only a limited number of Boolean functions can be expressed exclusively in terms of
exclusiveOR or equivalence operations. Nevertheless, these functions emerge quite often
during the design of digital systems. The two functions are particularly useful in
arithmetic operations and error detection and correction.
DESIGN OF COMBINATIONAL CIRCUITS
The design of combinational circuits starts from the verbal outline of the problem and
ends in a logic circuit diagram. The procedure involves the following steps:
1. State the given problem completely and exactly.
2. Interpret the problem and determine the available input variables and required output
variables.
3. Assign a letter symbol to each input and output variables.
4. Design the truth table that defines the required relations between inputs and outputs.
5. Obtain the simplified Boolean function for each output.
6. Draw the logic circuit diagram to implement the Boolean function.
The design procedure is illustrated below with the design of adder circuits because
addition is the most basic a...
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 Spring '14

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