Video: 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 𝑝 in the table? What is the value of π‘ž in the table?

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Video Transcript

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 π‘ž 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 π‘ž. The questions we’ve been asked are, what is the value of 𝑝 and what is the value of π‘ž?

Since 𝑝 and π‘ž 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 π‘ž 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 π‘ž 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 π‘ž in the table is zero.

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