Chapter 10.3 and 10.4: Combinatorial Circuits

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Chapter 10.3 and 10.4 Combinatorial Circuits

• Discrete Mathematical Structures
• Theory and Applications

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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 digitsx 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

• First mark the 1s that cannot be paired with any
  other 1. Put a circle around them.
• 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.
• 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.
• 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.
• 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.)
• 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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