In this explainer, we will learn how to determine the input and output of NOT gates in logic circuits and complete truth tables for NOT gates.
Recall that a logic gate is a device that takes one or more binary inputs and has one binary output. A binary signal has two possible values: 0 and 1.
There are other sets of terms we might use to express binary values, besides the numeric assignment 0 or 1. For instance, we may also see the values expressed as “false” and “true” or as “off” and “on.” Here, “false,” “off,” and 0 have the same value, as do “true,” “on,” and 1. The key point is that there are only two possible binary values, no matter what we call them, and the terminology we use just depends on the context we are working in. For instance, in an electrical circuit, we often use “off/on” to tell whether there is a current present in a circuit element.
Electronic devices like computers and smartphones process binary units of information by combining millions or billions of logic gates in specific ways. We will learn how NOT gates can be combined later in this explainer; for now, let us explore how NOT gates work.
The diagram above shows the symbol for a NOT gate. This symbol has features that are unique to NOT gates, which allows for easy and quick identification. For instance, NOT gates only have one input value, as shown by the single input line. This sets the NOT gate symbol apart from other logic gate types, which have more than one input. Here, the input is shown on the left and the output is on the right, which is made evident by the direction that the triangular “arrow” shape points. Further, the circle at the tip of the triangle represents an inversion—inverting a value either switches a 0 to a 1 or a 1 to a 0. This is the key function of the NOT gate: to take an input value, invert it, and pass the inverted value as the output.
Because there is only one input that can have two possible values, there are only two possible operations of the NOT gate, as shown in the diagrams below. We will use color to code the values, with red representing 0 and blue representing 1. Note that the dotted line extensions symbolize that the input and output lines continue in either direction, and that the two gates are separate.
The role of the NOT gate is to simply invert the input value. This can be quickly remembered using the following phrase: “The output value is not the input value.”
We can use a truth table to more formally show the functionality of a NOT gate by representing the possible combinations of input and output values by columns and rows. As shown below, there is one column for the input and one for the output. Further, since there is only one input that can have two possible values, there are two rows in the table.
Remember that the role of the NOT gate is to invert a value, which is made evident by the truth table; if the input is 0, the gate outputs a 1, as shown in the first row. Similarly, if a value of 1 is input, the gate outputs a 0, as shown in the bottom row.
The truth table reiterates the function of a NOT gate, which is worth stating formally.
Rule: NOT Gates
A NOT gate is a logic gate with one binary input and one binary output. The function of a NOT gate is to invert a value so that the input value is not the output.
Now that we have discussed the function of a NOT gate, let us work through a couple of examples.
Example 1: Evaluating the Output of NOT Gates
The diagram shows a NOT gate. If the input is 0, what will the output be?
Recall that a NOT gate takes an input value, inverts it, and passes the inverted value as an output.
Thus, since the input here is 0, we know that the gate will output a 1.
Example 2: Evaluating the Input of NOT Gates
The diagram shows a NOT gate. If the output is 0, what must the input be?
Recall that a NOT gate inverts its input, meaning that if the input is 1, the output will be 0. Further, if the input is 0, the output will be 1. In other words, the input value is not the output.
Since the output here is 0, we know that the gate must have an input of 1.
It is important to understand how to combine NOT gates. In this case, each individual gate behaves as we have seen so far, and the output of one gate gets passed along as the input for the next gate. Let us look at some examples where NOT gates are connected together.
Example 3: Evaluating the Output of Multiple NOT Gates Using Truth Tables
The diagram shows three NOT gates connected as part of a logic circuit. The truth table shows the two different possible inputs.
- What is the value of in the table?
- What is the value of in the table?
Here we have the truth table for three NOT gates combined in series. Each individual gate behaves as normal, simply inverting the input value so that the input is not the output.
To fill out the truth table, we will work through the gates one by one, determining their outputs in order.
The value appears as an output when a value of 0 is input. When we input a value of 0 to the first gate, it outputs a 1.
This value is passed along as the input for the second gate. As shown below, the second gate outputs a 0.
The third gate has an input of 0, so it outputs a 1.
Thus, we know that if this combination of NOT gates has an input value of 0, the final output value will be 1.
Therefore the value of is 1.
Now let us consider the three NOT gates combined in series with an initial input value of 1, which is shown in the diagram below. Like before, the original input value is inverted three times, so the final output of this series combination of three NOT gates is 0.
Thus, when this combination of NOT gates has an initial input value of 1, it has a final output value of 0.
Therefore, the value of in the table is 0.
Following the same pattern that we just explored, we can imagine what would happen for any other odd-numbered series combination of NOT gates. Each NOT gate takes turn alternating values, so the inversions of any two consecutive NOT gates end up effectively canceling one another. Thus, we know that that any odd-numbered series combination of NOT gates will have alternate input and output values. In other words, if we input a value into an odd-numbered series combination of NOT gates, the final output will have the same value as a single NOT gate would have. This is worth restating.
Rule: Odd-Numbered Series Connection of NOT Gates
Any odd number of NOT gates connected in series will produce the same output as a single NOT gate.
Now let us explore another combination of NOT gates.
Example 4: Evaluating the Output of Multiple NOT Gates
Four NOT gates are connected together in series. If the input of the first NOT gate is 0, what will the output of the final NOT gate be?
Recall that the job of a NOT gate is to invert a value so that the input value is not the output.
We will work through the gates one by one, beginning with the first NOT gate. Here we input a value of 0, so the first gate outputs a 1. This value of 1 becomes the input for the second gate.
We continue this pattern of inverting values at each gate until we reach the end, as summarized by the diagram below.
Thus, if we input a value of 0 to a combination of four NOT gates, the output has a value of 0.
This example was similar to the previous one, but this time we saw a trend for combining even numbers of NOT gates. When we look at any pair of NOT gates, we can see that the two gates take an input and invert it twice. Applying this pattern to any multiple of two gates, we know that any even-numbered combination of NOT gates will output the same value that is input. This is worth restating.
Rule: Even-Numbered Series Connection of NOT Gates
Any even-numbered series combination of NOT gates will output the same value that is originally input.
Let us finish by summarizing a few important concepts.
- A NOT gate is a type of logic gate with one binary input and one binary output.
- The symbol representation for a NOT gate is shown below.
- The key function of a NOT gate is to invert a value so that a 0 input gives a 1 output or a 1 input gives a 0 output.
- The symbol for a NOT gate refers to its function: one input value is passed along in the direction that the arrow “points,” and the value is inverted (as represented by the inversion circle at the tip of the arrow) and is then passed as the output.
- We can use a truth table to formally represent the functionality of one or more NOT gates.
- Any even-numbered series combination of NOT gates will output the same value that is originally input.
- Any odd number of NOT gates connected in series will produce the same output as a single NOT gate.