1. Chapter 10.3 and 10.4: Combinatorial Circuits
Discrete Mathematical Structures:
Theory and Applications

2. Learning Objectives
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. Logical Gates and Combinatorial Circuits
Discrete Mathematical Structures: Theory and Applications

4. Logical Gates and Combinatorial Circuits
Discrete Mathematical Structures: Theory and Applications

5. Logical Gates and Combinatorial Circuits
Discrete Mathematical Structures: Theory and Applications

6. 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.
Discrete Mathematical Structures: Theory and Applications

7. Logical Gates and Combinatorial Circuits
Discrete Mathematical Structures: Theory and Applications

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20. Examples 2 and 3, p. 714
Discrete Mathematical Structures: Theory and Applications

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29. Logical Gates and Combinatorial Circuits
The diagram in Figure represents a circuit with more than one output.
Discrete Mathematical Structures: Theory and Applications

30. 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.
Discrete Mathematical Structures: Theory and Applications

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32. Discrete Mathematical Structures: Theory and Applications

33. 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

34. 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

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37. 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

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40. 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

41. Discrete Mathematical Structures: Theory and Applications