Video: Using NOR gates to Make Other Types of Logic gates

The diagram shows 2 NOR gates connected as part of a logic circuit. Use a truth table to work out what type of logic gate this arrangement is equivalent to.

07:10

Video Transcript

The diagram shows two NOR gates connected as part of a logic circuit. Use a truth table to work out what type of logic gate this arrangement is equivalent to?

Alright, so, in the diagram that we’ve been given, we can see that there are two NOR gates connected with each other. In the first NOR gate, we’ve got two independent inputs, input A and input B. And as usual, a NOR gate has one output. However, that one output is then being split to give us the two inputs of the second NOR gate. In other words, the two inputs to the second NOR gate will always have the same values as each other, and those input values will be the same as the output value from the first NOR gate.

Now, we’ve been asked to use a truth table to work out what type of logic gate this entire arrangement is equivalent to. But to work this out, we should first start by recalling the truth table for a single NOR gate. So, here’s the truth table for a single NOR gate, assuming two independent inputs, which here we’ve called 𝛼 and 𝛽.

Now, the reason that a NOR gate is called a NOR gate is because we can think of this type of gate as a NOT OR gate. Now, an OR gate is a gate which will return an output of one if either one of the inputs has a value of one. In other words, the truth table for an OR gate looks something like this. When both of the inputs are zero, then the output is Zero. However, in the second row, we can see that input 𝛼 is zero but input 𝛽 is one. And this means that one of the inputs is one, and so the output is going to be one as well.

Similarly, in the third row, one of the inputs is one, and so the output is going to be one as well. And in the final row, both of the inputs are one, and so the output is going to be one also. So, basically, for an OR gate, whenever either one of the inputs or the other is one, then the output is going to be one. That’s why it’s called an OR gate.

Coming back to our NOR gate though, we said that it’s a NOT OR gate. And a NOT gate, otherwise known as an inverter, will take the values from the OR gate and invert them to give us the NOR gate. In other words, all the zeros become ones, and all the ones become zeros, as we can see here. So, anyway, this is the truth table for a NOR gate, assuming independent inputs. In other words, this assumes that we can change the values of 𝛼 and 𝛽 individually. So, to do this, let’s create our own truth table that will represent this setup here.

Let’s start by creating two columns, one for each input that we’ve been given, input A and input B. And because they’re separate inputs, we can vary them independently. Now, whatever the values of input A and B, those will be processed by our first NOR gate. And it will result in an output value from that first NOR gate. So, to represent this, we’re going to add an extra column to our truth table. However, we can’t call this column output because the word output is actually referring to the output of the second NOR gate, or in other words the output of the entire arrangement.

So, let’s call the output of the first NOR gate Ω. And then, we can see that this output then splits into the two inputs for the second NOR gate. Now, we can call these inputs anything that we want. Let’s call them 𝛾 and 𝛿 and give each one their own column in the truth table as well. And finally, we see that the second NOR gate will process the inputs 𝛾 and 𝛿 and give us an output value. So, we can add an output column to our truth table. Now, let’s start filling in this truth table.

We know that we can vary the values of input A and input B. And we need to work out what this arrangement does to those input values and what output that gives us. So, let’s start by setting input A to zero and input B to zero, just like we would in any one of these truths tables. Now, if input A is zero and input B is zero, then this is being processed by our first NOR gate. And so, we can use a NOR gate truth table to tell us that the value of Ω, which is the output of the first NOR gate, is actually going to be one. Hence, we can say that the value of Ω in this instance is one.

But then, as we’ve already seen the output Ω feeds into the inputs 𝛾 and 𝛿. In other words, the inputs 𝛾 and 𝛿 are going to have the same values as Ω always. Therefore, in this case, the value of input 𝛾 is one and the value of input 𝛿 is one. And then, 𝛾 and 𝛿 feed into a second NOR gate. So, we need to find the row in this truth table that corresponds to the two inputs of the NOR gate being both ones.

And that row is the final rule here. We can see when the value of the first input is one, and the value of the second input is one, the output of that NOR gate is going to be zero. And as it turns out, the output of this second NOR gate is the same thing as the output of the entire arrangement. And so, we can say that the output value is going to be zero. So, we’ve now filled in one row of this truth table.

Let’s now change the value of input B. Let’s see that input A stays zero and input B now becomes one. Well, in that case, we can see what happens with the first NOR gate using our truth table here once again, we can see that if the first input is zero, and the second input is one, then the output of this NOR gate is going to be zero. In other words, our value for Ω is zero.

And once again, Ω feeds into input 𝛾 and input 𝛿. And so, input 𝛾 and 𝛿 are going to have the same values as Ω, both zeros. Then, we come back to our NOR gate’s truth table to see that when both the inputs are zero, then the output of this NOR gate is going to be one. And so, our output is going to be one here.

Now, we can repeat this whole process for when input A is one and input B is zero. Looking at the truth table, we see that the output Ω is going to be zero. Therefore, 𝛾 and 𝛿 are going to be zero. And then, when both inputs to the second NOR gate are zero, then the output of the second NOR gate is going to be one.

And finally, let’s see what happens when we set both input A and input B to be one. In that situation, we’re looking at the final row of the table. When both inputs to the first NOR gate are one, then the output Ω is going to be zero. And hence, of course, 𝛾 and 𝛿 are going to be zero. And so, when both inputs to the second NOR gate are zero, then the output to the second NOR gate is going to be one.

So, at this point we filled in our truth table. We’ve accounted for all possible combinations of input A and input B. And we’ve figured out what value the output is going to have in each case. So, at this point, we can take our truth table and simplify it slightly. We can now think of this whole setup as one logic gate. What happens within this logic gate is irrelevant to us. In other words, we don’t anymore care about the values of Ω, 𝛾, or 𝛿. All we care about is what we put in as input A and B and what comes out as the output.

Therefore, we can get rid of these three columns representing Ω, 𝛾, and 𝛿 and just keep the columns for input A, input B, and the output. When we do this, we’re left with our final truth table, which assumes that we’re taking these inputs and sending them into a single logic gate, or at least something that behaves like a single logic gate. So, let’s try and work out what logic gate this truth table represents.

Well, actually, if we look closely, we can realize that we’ve already seen this truth table down here. This is the truth table of an OR gate. For example, we can see that when both inputs are zero, then the output is zero, just like we see here. When the first input is zero and the second input is one, then the output is one. When the first input is one and the second input is zero, then the output is one. Once again, this checks out. And when both inputs are one, the output is one as well.

In other words then, this whole setup that we’ve got here, where we take a NOR gate and feed the output of that NOR gate into both the inputs of a second NOR gate, we see that this whole system then behaves like an OR gate. So, at this point, we found the answer to our question. We’ve used the truth table to work out that the type of logic gate this arrangement is equivalent to is an OR gate.

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