Chapter 10.3 and 10.4:
Combinatorial Circuits
Discrete Mathematical Structures:
Theory and Applications
Learning Objectives
Explore the application of Boolean algebra in the
design of electronic circuits
Learn the application of Boolean algebra in
switching circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
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Logical Gates and Combinatorial Circuits
In circuitry theory, NOT, AND, and OR gates are the
basic gates. Any circuit can be designed using these
gates. The circuits designed depend only on the inputs,
not on the output. In other words, these circuits have
no memory. Also these circuits are called
combinatorial circuits.
The symbols NOT gate, AND gate, and OR gate are
also considered as basic circuit symbols, which are
used to build general circuits. The word circuit instead
of symbol is also used.
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Logical Gates and Combinatorial Circuits
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Examples 2 and 3, p. 714
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Logical Gates and Combinatorial Circuits
 The diagram in Figure 12.32
represents a circuit with
more than one output.
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A half adder is a circuit that accepts as input two binary digits
x and y,
and produces as output the sum bit s and the carry bit c.
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Logical Gates and Combinatorial Circuits
 A NOT gate can be
implemented using a
NAND gate (see
Figure 12.36(a)).
 An AND gate can be
implemented using
NAND gates (see
Figure 12.36(b)).
 An OR gate can be
implemented using
NAND gates (see
Figure12.36(c)).
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Logical Gates and Combinatorial Circuits
Any circuit which is designed by using NOT, AND,
and OR gates can also be designed using only
NAND gates.
Any circuit which is designed by using NOT, AND,
and OR gates can also be designed using only NOR
gates.
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Logical Gates and Combinatorial Circuits
 The Karnaugh map, or K-map for short, can be used to
minimize a sum-of-product Boolean expression.
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Logical Gates and Combinatorial Circuits
1.
First mark the 1s that cannot be paired with any other 1. Put a circle around them.
2.
Next, from the remaining 1s, find the 1s that can be combined into two square
blocks, i.e., 1 x 2 or 2 x 1 blocks, and in only one way.
3.
Next, from the remaining 1s, find the 1s that can be combined into four square
blocks, i.e., 2 x 2, 1 x 4, or 4 x 1 blocks, and in only one way.
4.
Next, from the remaining 1s, find the 1s that can be combined into eight square
blocks, i.e., 2 x 4 or 4 x 2 blocks, and in only one way.
5.
Next, from the remaining 1s, find the 1s that can be combined into 16 square
blocks, i.e., a 4 x 4 block. (Note that this could happen only for Boolean
expressions involving four variables.)
6.
Finally, look at the remaining 1s, i.e., the 1s that have not been grouped with any
other 1. Find the largest blocks that include them.
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