LOGIC SYMBOL

The standard symbol for the AND gate is shown in figure 2-2. Variations of this standard symbol may be encountered. These variations become necessary to illustrate that an AND gate may have more than one input.

Figure 2-2. - AND gate.

AND GATE OPERATION

We can demonstrate the operation of the AND gate with a simple circuit that has two switches in series as shown in figure 2-3. You can see that both switches would have to be closed at the same time to light the lamp (view A). Any other combination of switch positions (view B) would result in an open circuit and the lamp would not light (logic 0).

Figure 2-3. - AND gate equivalent circuit: A. Logic 1 state; B.

Logic 0 state.

Now look at figure 2-4. Signal A is applied to one input of the AND gate and signal B to the other. At time T₀, both inputs are LOW (logic 0) and f is LOW. At T₁, A goes HIGH (logic 1); B remains LOW; and as a result, f remains LOW. At T₂, A goes LOW and B goes HIGH; f, however, is still LOW, because the proper input conditions have not been satisfied (A and B both HIGH at the same time). At T₄, both A and B are HIGH. As a result, f is HIGH. The input requirements have been satisfied, so the output is HIGH (logic 1).

Figure 2-4. - AND gate input and output signals.

TRUTH TABLE

Now let's refer to figure 2-5. As you can see, a Truth Table and a Table of Combinations are shown. The latter is a deviation of the Truth Table. It uses the HIGH and LOW logic levels to depict the gate's inputs and resultant output combinations rather than the 1 and 0 logic states. By comparing the inputs and outputs of the two tables, you see how one can easily be converted to the other (remember, 1 = HIGH and 0 = LOW). The Table of Combinations is shown here only to familiarize you with its existence, it will not be seen again in this book. As we mentioned earlier, the Truth Table is a chart that shows all possible combinations of inputs and the resulting outputs. Compare the AND gate Truth Table (figure 2-5) with the input signals shown in figure 2-4.

Figure 2-5. - AND gate logic symbol, Truth Table, and Table of Combinations.

The first combination (A = 0, B = 0) corresponds to T₀ in figure 2-4; the second to T₁; the third to T₂; and the last to T₄. When constructing a Truth Table, you must include all possible combinations of the inputs, including the all 0s combination.

A Truth Table representing an AND gate with three inputs (X, Y, and Z) is shown below. Remember that the two-input AND gate has four possible combinations, with only one of those combinations providing a HIGH output. An AND gate with three inputs has eight possible combinations, again with only one combination providing a HIGH output. Make sure you include all possible combinations. To check if you have all combinations, raise 2 to the power equal to the number of input variables. This will give you the total number of possible combinations. For example:

EXAMPLE 1-AB = 2² = 4 combinations

EXAMPLE 2-XYZ = 2³ = 8 combinations

f = XYZ

As with all AND gates, all the inputs must be HIGH at the same time to produce a HIGH output. Don't be confused if the complement of a variable is used as an input. When a complement is indicated as an input to an AND gate, it must also be HIGH to satisfy the input requirements of the gate. The Boolean expression for the output is formulated based on the TRUE inputs that give a TRUE output. Here is an adage that might help you better understand the AND gate:

In order to produce a 1 output, all the inputs must be 1. If any or all of the inputs is/are 0, then the output will be 0.

Referring to the following examples should help you cement this concept in your mind. Remember, the inputs, whether the original variable or the complement must be high in order for the output to be high. The three examples given are all AND gates with two inputs. Keep in mind the Boolean expression for the output is the result of all the inputs being HIGH.

You will soon be able to recognize the Truth Table for the other types of logic gates without having to look at the logic symbol.

Q.1 What is defined as "the science of reasoning?"
Q.2 With regard to computer logic circuits, what is meant by "complement?"
Q.3 What are the complements of the following terms?

Q.4 If logic 1 = -5 vdc and logic 0 = -10 vdc, what logic polarity is being used?
Q.5 If logic 1 = +2 vdc and logic 0 = -2 vdc, what logic polarity is being used?
Q.6 If logic 1 = -5 vdc and logic 0 = 0 vdc, what logic polarity is being used?
Q.7 What is the Boolean expression for the output of an AND gate that has R and S as inputs?
Q.8 What must be the logic state of R and S to produce the TRUE output?
Q.9 How many input combinations exist for a four-input AND gate?