Lecture 21: Combinatorial Circuits II

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Lecture 21 Combinatorial Circuits II

• Discrete Mathematical Structures
• Theory and Applications

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Learning Objectives

• Learn about Boolean expressions
• Become aware of the basic properties of Boolean
  algebra
• 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

• The diagram in Figure 12.32 represents a circuit
  with more than one output.

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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

• 1s should be circled in the largest group of a
  power of 2 (1,2,4,8, etc.) to which they belong.
• There are six steps to be followed when deciding
  how to circle blocks of 1s.

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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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