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# THE NOR GATE by Isai Damier

The NOR gate is an electronic circuit that performs joint denial. The output of the NOR Boolean operator is true only when all the inputs are false. Otherwise, the output is false.

TRUTH TABLE
Input Output Comment
A B Out
0 0 1 High
0 1 0 Low
1 0 0 Low
1 1 0 Low
The NOR gate is known as the joint denial operator because to say neither A nor B is the same as to say not A and not B (DeMorgan’s theorem). For example, to say neither enjoyment nor sorrow is our destined end or way is the same as to say not enjoyment and not sorrow is our destined end or way (Longfellow).

Consider the statement if Dave passes neither calculus nor physics he will lose his scholarship. In analyzing this statement we observe that Dave can suffer four possible circumstances, as shown in Table 1.

PossibilitiesPass calculusPass physicsCOMMENT
Case 1 FALSE FALSE Pass neither
Case 2 FALSE TRUE Pass physics
Case 3 TRUE FALSE Pass calculus
Case 4 TRUE TRUE Pass both

Table 1: possible input conditions

Per the statement, the importance of the possible circumstances lies in their outcome: whether Dave will lose his scholarship or not. Hence, we continue our analysis by assigning outcome to each circumstance. We show the result in Table 2. Observe that Dave only loses his scholarship when he is jointly denied a passing grade in calculus and in physics. Under any other circumstance, Dave gets to keep his scholarship.

Pass calculusPass physicsLose scholarshipCOMMENT
FALSE FALSE TRUE Poor Dave. No money now.
FALSE TRUE FALSE Dave is still in the game!
TRUE FALSE FALSE Dave is still in the game!
TRUE TRUE FALSE Way to go, Dave!

Table 2: Truth Table of complete evaluation

An important discovery about the word NOR is that it can express all logical operations. In other words, if you become good at using the word NOR, then you will never need to use any of the operators AND, OR, NOT to adequately express yourself. Your friends may find you weird, but you will make logical sense. The following two accounts are logically equivalent.

• My friend and I did not eat egg or sausage for breakfast.
• Neither did nor my friend nor I eat neither egg nor sausage for breakfast.

You should try that trick on a date. You never know that your date may find your strange way of talking cute.

Because all the other logic gates can be constructed using the NOR gate, in digital systems we say the NOR gate is universal. Mathematicians use the term functionally complete and linguists say expressively adequate. It is typical in engineering to use 1 instead of TRUE and 0 instead of FALSE. Therefore we rewrite the data from Table 2 in Table 3 accordingly.

Pass calculusPass physicsLose scholarshipCOMMENT
0 0 1 Poor Dave. No money now.
0 1 0 Dave is still in the game!
1 0 0 Dave is still in the game!
1 1 0 Way to go, Dave!

Table 3: Truth Table of complete evaluation

For greater detail on the logic significance of the word NOR, read the Boolean Algebra article.

#### Physical Implementation

In order to apply the principles of Boolean algebra to create real machines that can think and make decisions, we have had to find ways to physically implement the logic operators AND, OR, NOT, etc. To that end, modern day engineering uses transistor networks called logic gates. Hence, a logic gate is actually a group of transistors so arranged as to behave as a Boolean operator.

From a circuit complexity perspective, the most basic logic gate is the NOT gate (aka the Inverter). The NOT gate is made of two transistors, as shown in Figure 1. The next most basic logic gate is the NOR gate, which is effectively two Inverters as shown in Figure 2. Hence, we only need four transistors to build a NOR gate.

Figure 1: Interactive transistor circuit of the NOT logic operator

Figure 2: Interactive transistor circuit of the NOR logic operator

#### Transistors

The use of transistors to build logic gates is quite modern. Before transistors we used other devices, such as vacuum tubes (aka thermionic valves). And very soon we may use DNA, or some other abundant material. There are many types of transistors. Our circuits in figures 1 and 2, for example, use complementary metal–oxide semiconductor (CMOS) technology. Our choice of CMOS is arbitrarily based on the fact that CMOS is by far the dominant technology in use today. The dominance is due to how well CMOS performs in all the important categories: fabrication cost, packing density, loading capacity (i.e. fan-out), operational speed (i.e. propagation delay), noise margin, and power dissipation (i.e. green technology).

There is of course more to transistors than can be presented here; especially since transistors are used for more than just digital systems. And so we refer you to any good micro–electronics textbook.

#### Alternate Design

Below we show two additional typical constructions of the NOR gate. Each of the constructions presents specific conveniences to designers. If you are very new to digital systems design, you may not understand the importance of the figures below. Still, we include them in this article for the people who may need them.

Interactive NOR gate implemented with three NAND gates digital logic circuit with boolean algebra equation and truth table
Figure 3: NOR gate implemented with three NAND gates

Interactive NOR gate implemented with a 2X1 Mux digital logic circuit with boolean algebra equation and truth table
Figure 4: NOR gate implemented with a 2X1 Mux

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