Module 13 - Introduction to Number Systems and Logic
Navy Electricity and Electronics Training Series (NEETS)
Chapter 2:  Pages 2-21 through 2-30

Figure 2-22. - NOR gate with one inverted input.

Table 2-4 illustrates and, NOR, Nand, and OR gate combinations that produce the same output. You can see by the table that there is more than one way to achieve a desired output. Although the gates have only two inputs, the table can be extended to more than two inputs.

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Table 2-4. - Equivalent and and NOR, Nand and OR Gates

Q24.   What is the output Boolean expression for an and gate with a and B as inputs when the B input is inverted?

Q25.   What is the equivalent logic gate of a two-input Nand gate with both inputs inverted?

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Q26.   What is the output Boolean expression for the following gates?

LOGIC GATES IN Combination

When you look at logic circuit diagrams for digital equipment, you are not going to see just a single gate, but many combinations of gates. At first it may seem confusing and complex. If you interpret one gate at a time, you can work your way through any network. In this section, we will analyze several combinations of gates and then provide you with some practice problems.

Figure 2-23 (view A) shows a simple combination of and gates. The outputs of gates 1 and 2 are the inputs to gate 3. You already know that both inputs to an and gate must be High at the same time in order to produce a High output.

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Figure 2-23. - Logic gate combinations: A. Simple combination of and gates; B. Simple combination of and gates and OR gate.

The output Boolean expression of gate 1 is RS, and the output expression of gate 2 is TV. These two output expressions become the inputs to gate 3. Remember, the output Boolean expression is the result of the inputs, in this case (RS)(TV); spoken "quantity R and S and quantity T and V."

In view B we have changed gate 3 to an OR gate. The outputs of gates 1 and 2 remain the same but the output of gate 3 changes as you would expect. The output of gate 3 is now (RS)+(TV); spoken "quantity R and S OR quantity T and V."

In figure 2-24 (view A), the outputs of two OR gates are being applied as the input to third OR gate. The output for gate 1 is R+S, and the output for gate 2 is T+V. With these inputs, the output expression of gate 3 is (R+S)+(T+V).

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Figure 2-24. - Logic gate combinations: A. Simple combination of OR gates; B. Simple combination of OR gates and and gate; C. Output expression without the parentheses.

In view B, gate 3 has been changed to an and gate. The outputs of gates 1 and 2 do not change, but the output expression of gate 3 does. In this case, the gate 3 output expression is (R+S)(T+V). This expression is spoken, "quantity R OR S and quantity T OR V." The parentheses are used to separate the input terms and to indicate the and function. Without the parentheses the output expression would read R+ST+V, which is representative of the circuit in view C. As you can see, this is not the same circuit as the one depicted in view B. It is very important that the Boolean expressions be written and spoken correctly.

The Truth Table for the output expression of gate 3 (view B) will help you better understand the output. When studying this Truth Table, notice that the only time f is High (logic 1) is when either or both R and S and either or both T and V are High (logic 1).

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 Now let's determine the output expression for the NOR gate in figure 2-25. First write the outputs of gates 1, 2, and 3:

Figure 2-25. - Logic gate combinations.

Since all three outputs are applied to gate 4, proceed as you would for any NOR gate. We separate each input to gate 4 with an OR sign (+) and then place a vinculum over the entire expression. The output expression of gate 4 is:

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When you are trying to determine the outputs of logic gates in combination, take them one gate at a time!

Now write the output expressions for the following logic gate combinations:

Q27.

Q28.

Q29.

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Q30.

Q31.

Q32.

BOOLEAN ALGEBRA

Boolean logic, or Boolean algebra as it is called today, was developed by an English mathematician, George Boole, in the 19th century. He based his concepts on the assumption that most quantities have two possible conditions ¾ TRUE and FALSE. This is the same theory you were introduced to at the

beginning of this chapter.

Throughout our discussions of fundamental logic gates, we have mentioned Boolean expressions. a Boolean expression is nothing more than a description of the input conditions necessary to get the desired output. These expressions are based on Boole's laws and theorems.

PURPOSE

Boolean algebra is used primarily by design engineers. Using this system, they are able to arrange logic gates to accomplish desired tasks. Boolean algebra also enables the engineers to achieve the desired output by using the fewest number of logic gates. Since space, weight, and cost are important factors in the design of equipment, you would usually want to use as few parts as possible.

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Figure 2-26 (view A), shows a rather complex series of gates. Through proper application of Boolean algebra, the circuit can be simplified to the single OR gate shown in view B. Figure 2-27 shows the simplification process and the Boolean laws and theorem used to accomplish it.

 Figure 2-26. - Logic simplification: A. Complex series of gates; B. Simplified single OR gate.

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Figure 2-27. - Logic circuit simplification process.

LAWS and THEOREMS

Each of the laws and theorems of Boolean algebra, along with a simple explanation, is listed below.

LAW of IDENTITY - a term that is TRUE in one part of an expression will be TRUE in all parts of the expression (A = a or a = A).

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