Lecture 21: Combinatorial Circuits II Discrete Mathematical Structures: Theory and Applications

Discrete Mathematical Structures: Theory and Applications 2 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

Discrete Mathematical Structures: Theory and Applications 3

4

5

6

7

8

9 Logical Gates and Combinatorial Circuits  The diagram in Figure represents a circuit with more than one output.

Discrete Mathematical Structures: Theory and Applications 10

Discrete Mathematical Structures: Theory and Applications 11

Discrete Mathematical Structures: Theory and Applications 12

Discrete Mathematical Structures: Theory and Applications 13 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)).

Discrete Mathematical Structures: Theory and Applications 14 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.

Discrete Mathematical Structures: Theory and Applications 15

Discrete Mathematical Structures: Theory and Applications 16

Discrete Mathematical Structures: Theory and Applications 17 Logical Gates and Combinatorial Circuits  The Karnaugh map, or K-map for short, can be used to minimize a sum-of-product Boolean expression.

Discrete Mathematical Structures: Theory and Applications 18

Discrete Mathematical Structures: Theory and Applications 19

Discrete Mathematical Structures: Theory and Applications 20

Discrete Mathematical Structures: Theory and Applications 21

Discrete Mathematical Structures: Theory and Applications 22 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.

Discrete Mathematical Structures: Theory and Applications 23 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.

Discrete Mathematical Structures: Theory and Applications 24