Video: NOT Gates

In this video, we will learn how to determine the input and output of NOT gates in logic circuits and complete truth tables for NOT gates.

15:55

Video Transcript

In this video, we’re going to be looking at a type of logic gate known as a NOT gate. A logic gate is a component with one or more inputs and one output, each of which can take one of two values, zero or one. A logic gate will determine the value of its output based on the input values it receives. Now, before we look at the specific properties of a NOT gate, let’s first discuss some of the terminology that we use when we’re talking about logic gates.

As we’ve already mentioned, the inputs and output of a logic gate can each take one of two values. And these two values are often represented as either zero or one. So the input can take a value of zero or one and so can the output. However, there are many different ways of representing the two values that an input or output can take. For example, instead of saying that an output is set to zero, we may say that it’s set to false. And conversely, instead of saying that an input or output is set to one, we could say that it’s set to true. Alternatively, instead of using zero and one, or false and true, we could use off and on.

For example, let’s say that our input is set to zero and our output is set to one. It would then be equivalent to say that the input was set to false and the output was set to true or to say that the input was off and the output was on. It doesn’t matter which set of terminology we use as they’re all just different ways of representing the two possible states of an input or an output. But we do need to be aware of the different words we can use to describe them.

It’s worth noting that we use the off and on terminology most commonly when we’re talking about electrical circuits. For example, let’s consider a logic gate in an electrical circuit. So here’s our logic gate, specifically a NOT gate, and we have an input on the left and an output on the right. These dotted lines just show us that the circuit extends in this direction and this direction. In this context, the value of an input describes whether or not a voltage is being applied to it. So if there were no voltage being applied to the input, then we could say that it’s off. And this is equivalent to saying that it has a value of zero or false. If we were to then apply a voltage to the input, then its value would change to on. Or equivalently, we could say that its value has now changed to one or true.

Similarly, the value of our output describes whether or not the logic gate is supplying a voltage to it. So if the logic gate is providing a voltage to the output, we can say that its value is on or one or true. Or if the logic gate is not providing a voltage to the output, then we can say that its value is off or zero or false. So now that we started looking at a NOT gate, let’s consider it in a bit more detail. The first thing to know is that this is the symbol that we use to represent a NOT gate.

In this diagram, the input is on the left and the output is on the right. We can see that the NOT gate has only one input. And like all logic gates, it has one output. As we mentioned earlier, logic gates will give a certain output depending on the inputs that are provided to it. In the case of a NOT gate, the relationship between the input and the output is pretty simple. If the input is zero, then the output will be one or if the input is one, then the output will be zero. And that’s basically all there is to it.

One useful way of representing how the output of a logic eight depends on its inputs is by using a truth table. In a truth table, we have one column showing the possible values of our output and another column for each input. So in the case of our NOT gate, because there’s only one input, we only have one input column in our truth table. We also have an output column that shows us the corresponding output for each of the inputs shown in the input column. In the input column, we can write down the two possible inputs to a NOT gate, zero or one. Now, in the output column, we can write down the output values that a NOT gate will give us for each of the possible inputs. When the input of a NOT gate is zero, we find that the output is one. And when the input to a NOT gate is one, we find that we get an output of zero.

And that’s it. We now have the completed truth table for a NOT gate. This truth table shows us why we call the NOT gate a NOT gate. The output that the gate produces is always the opposite of the input. In other words, the output value is not the input value. The symbol that we use to represent a NOT gate is actually related to the way that it behaves. This part of the symbol represents an arrow pointing from the input to the output, which on its own would suggest that whatever value we put into the input would then be passed to the output.

However, the symbol for a NOT gate also includes this circle. This is also known as an inverting circle. And it’s used in logic gate symbols to represent inverting a value, which means if it’s a zero, changing it to a one and if it’s a one, changing it to a zero. So the symbol for a NOT gate actually tells us what it does. It takes a value from the input, passes it in the direction of the arrow, and then inverts it so that the output value will be the inverse or opposite of the input value.

So now that we’ve looked at what a NOT gate does, let’s solidify our understanding by looking at a NOT gate in action.

Let’s consider a simple situation where the output of a NOT gate is connected to a lamp. Once again, we’re using dotted lines to show that the circuit extends in this direction and in this direction. And really, we just do this so that we don’t have to draw in the rest of the circuit. Let’s start by thinking about what happens when the input of our NOT gate is set to zero. Because we’re dealing with an electrical circuit, it would also make sense to say that our input is set to off. But just to keep things simple in this explanation, we’ll stick to zeros and ones. A truth table tells us that when the input of a NOT gate is set to zero, then the output will take a value of one.

In a circuit context, an output value of one means that charge will flow through the output. Hence, there is a current through the lamp, causing it to light up. Conversely, if we set the input to one, then we can see from the bottom row of our truth table that the output will take a value of zero. In a circuit context, this means that no current is flowing through the output, which means no current is flowing through the lamp. So it won’t light up. Now, this isn’t actually a very useful application for a NOT gate as changing our input state will just switch the light on and off, which would still be true if we didn’t have a logic gate at all.

However, this doesn’t mean that NOT gates aren’t useful in real life. In fact, we need them for building most digital circuits. And any smart phone or computer relies on millions or billions of NOT gates in order to work properly. Now, let’s take a look at what happens when we connect the output of one NOT gate to the input of another. Let’s call these two NOT gates A and B. We can see that the output of NOT gate A is connected directly to the input of NOT gate B. This means that whatever value is output from NOT gate A will become the input of NOT gate B.

Let’s see what happens if, for example, the input value of NOT gate A is zero. Our truth table for a NOT gate shows us that, with an input value of zero, a NOT gate will produce an output value of one. So the output value of NOT gate A is one. And because this is connected to NOT gate B, that means that one is also the input value for NOT gate B. Once again, the truth table shows us what happens at this NOT gate. With an input value of one, the output value will be zero. So the output value of NOT gate B will be zero. So we can see that, in this case, the final output value is the same as the initial input value.

When we have several NOT gates connected together like this, we can think of the series of input and output values as being like a signal passing from left to right. The signal is changed or inverted every time it goes through a NOT gate, such that if it were initially a zero, it becomes a one. And if it were initially a one, it becomes a zero. Now, let’s see what these two connected NOT gates do when our input to NOT gate A is a one. Well, we know that each NOT gate inverts the signal. So NOT gate A will change its input of one into an output of zero. This then becomes the input for NOT gate B. So NOT gate B will invert its input of zero into an output of one.

This result confirms the fact that when we connect two NOT gates together like this, the final output will always be the same as the initial input. We can keep using the same method for larger numbers of NOT gates connected together. For example, let’s imagine we had three NOT gates connected together, called A, B, and C. If we have an initial input signal of zero going into NOT gate A, then it will produce an output signal of one. This then becomes the input signal for NOT gate B. So B produces an output value of zero. And then this zero becomes the input value for C. So C has an output of one. What we find is that when we have three NOT gates connected together, the final output value will be the inverse or opposite of the initial input value.

More generally, we find that if we have an odd number of NOT gates connected in series, like we have in our diagram, then the output will be the inverse of the initial input. But for any even number of NOT gates connected together in series, the output will be the same as the initial input. Okay, so now that we’ve looked at NOT gates in some detail, let’s have a look at some example questions.

Which of the following symbols represents a NOT gate?

So in this question, we’ve been given four fairly similar-looking symbols. And we need to figure out which one of them is used to represent a NOT gate. We can actually work out the answer to this question by recalling how a NOT gate works. The NOT gate is a logic gate that gets its name from the fact that its output value is not the same as the input value. The inputs and outputs of logic gates can only take the values zero or one. This means that if the input value of a NOT gate is zero, then its output value must be one. Conversely, if the input value of a NOT gate is one, then the output value must be zero.

We can represent this information in a truth table, which shows us the outputs that are produced as a result of all of the possible inputs. This statement and our truth table both tell us something important about NOT gates, that is that they only have one input. And like all other logic gates, they also have one output. We can use this information to help us decide which of our answer options is correct. When we draw a logic gate symbol, we usually draw the inputs on the left and the outputs on the right. If we look at option A, we can see that it has one output, and it has two inputs represented by the two horizontal lines on the left. This means that the symbol shown in option A can’t possibly be a NOT gate because it has two inputs and a NOT gate only has one input.

If we look at the other available options, we can also see that options B and C have two inputs each, which means that neither B nor C can be the correct answer either. So by process of elimination, we’re just left with option D as it’s the only option we’ve been given that just has one input as well as one output. To help us recall the symbol that we use to represent a NOT gate, we should recall that the symbol we use actually represents the function of a NOT gate. This small circle in the symbol for a NOT gate is actually used in many different logic gate symbols, and it represents an inversion. In other words, it represents a zero being changed to a one or a one being changed to a zero.

The other part of the symbol for a NOT gate is just an arrow pointing from the input to the output. So the symbol for a NOT gate represents the fact that it takes a value from the input, passes it towards the output, and inverts it along the way. Meaning that the output value will be the opposite or inverse of the input value. Just as we’ve shown in our truth table for a NOT gate. So this helps us confirm that the correct symbol that represents a NOT gate is indeed option D.

Now that we’ve answered that question, let’s take a look at another one.

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 𝑝 in the table? What is the value of 𝑞 in the table?

So in this question, we’ve been given a diagram showing a logic circuit consisting of three NOT gates. A logic circuit is made by connecting logic gates together such that the output of a logic gate becomes one of the inputs of another logic gate. For example, if we give the NOT gates in our diagram the names A, B, and C, then we could say that the output of NOT gate A is the input of NOT gate B. And similarly, the output of NOT gate B is the input of NOT gate C. We’ve also been given a truth table, although a couple of things about this truth table might seem slightly unusual.

A truth table is used to show how different inputs or combinations of inputs produce certain outputs. And we often use them to show how a single logic gate behaves. However, in this case, we can see that the input and output referred to in the table are not just the input and output of a single logic gate. Instead, they’re the input and output of a logic circuit which contains three NOT gates. In addition to this, we can see that instead of our outputs being given as zeros and ones, which would be normal for a truth table, the possible outputs are given as 𝑝 and 𝑞. The questions we’ve been asked are, what is the value of 𝑝 and what is the value of 𝑞?

Since 𝑝 and 𝑞 are in the output row of our table, these questions are essentially asking us, what are the possible outputs of our logic circuit? Specifically, 𝑝 refers to the output when the input to our circuit is zero. And 𝑞 refers to the output when the input to our circuit is one. So to find 𝑝, we need to figure out what the output of our circuit would be when the input is zero. Well, if our initial input to our circuit is zero, that means that NOT gate A has a zero as its input. Let’s recall that the output of a NOT gate will always be the inverse of its input. In other words, if we input a zero into a NOT gate, then it will output a one. And if we input a one into a NOT gate, then it will output a zero.

So if our input into NOT gate A is a zero, then its output will be a one. And our logic circuit diagram shows us that the output of A becomes the input of gate B. Since B is a NOT gate as well, inputting a one will mean that it outputs a zero. And this then becomes the input to gate C. Once again, gate C is a NOT gate, too. So inputting a zero will mean that it outputs a one. And we’ve now reached the end of our diagram. So we’ve shown that when we input a zero into our circuit diagram, then the output labeled in the diagram will be one. And since 𝑝 represents the output of the circuit when the input is zero, that means that the value of 𝑝 in the table is one.

The value of 𝑞 in the table represents the output of the circuit when the input to the circuit is one. So let’s see what happens when we input one into our logic circuit. As before, each of the NOT gates will invert the input, meaning a one is changed into a zero or a zero would be changed into a one. Since NOT gate A has an input of one, that means it has an output of zero. This means that NOT gate B has an input of zero, so it has an output of one. And finally, if gate C has an input of one, then it has an output of zero, which means that the overall output of the circuit is zero. And so the value of 𝑞 in the table is zero.

Okay, so now we’ve looked at a couple of example questions. Let’s summarize what we’ve talked about in this lesson. Firstly, we’ve seen that a NOT gate is a logic gate with one binary input and one binary output, where the word binary simply means that it can take one of two values. So the input of a NOT gate can take a value of either zero or one and so can the output. Secondly, we saw that the output of a NOT gate is the inverse of the input, meaning that if the input is a zero, then the output will be a one. And if the input is a one, then the output will be a zero. And finally, we saw that NOT gates, along with other logic gates, are commonly used in the circuitry found in computers.

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