### Video Transcript

The diagram shows a logic circuit
consisting of multiple logic gates. The table shows the output for some
of the different possible combinations of inputs.

So here, we have a circuit diagram
showing us three logic gates combined together to form a logic circuit. And here we have a table showing
some possible combinations of values for inputs A, B, C, and D and the value for the
output in each case.

Now, the first question asks us
what is the value of ๐ in the table?

And so if we look in the table, we
can see that itโs not fully populated with zeros and ones. In fact, for some rows, the output
values have been labelled as ๐, ๐, ๐, and ๐ . And the first question is asking us
to find the value of ๐. So to do this, the first thing that
weโre going to do is to identify the logic gates that we find in this logic
circuit. We can see that there are three
different logic gates but that two of them are identical. This logic gate has the same symbol
as this logic gate. Now we can recall, based on their
symbols, that these logic gates are in fact OR gates and that the other logic gate,
this one here, is an AND gate. And now, once weโve identified
this, itโs worth recalling the truth tables for OR and AND gates. So hereโs the truth table for an OR
gate. And hereโs the truth table for an
AND gate. Itโs worth noting that weโve used
๐ผ and ๐ฝ in both cases to represent the inputs and ๐บ to represent the outputs of
these gates. So that we avoid confusion with
these inputs and this output which has already been labelled in our diagram.

Now, we can see that, for an OR
gate, the output is set to one if either input ๐ผ or input ๐ฝ is set to one. And for an AND gate, the output is
only set to one if both input ๐ผ and input ๐ฝ are set to one. Now, coming back to our logic
circuit diagram, we can see that the output of the first OR gate becomes one of the
inputs to the AND gate. And the same is true for the output
of the second OR gate. It becomes the second input to the
AND gate. And so it might be worth extending
our truth table here slightly. Specifically, it might be worth
calling the output of the first OR gate position E and the output of the second OR
gate position F. Both E and F then go on to become
the inputs to the AND gate. And it might be worth tracking
whatโs happening to them, at least in the intermediate stages by adding two extra
columns to our truth table. We can delete these once weโve
completed the truth table. But for now, itโll be useful to
have them there.

So with that being said, if weโre
trying to calculate the value of ๐, then this corresponds to when input A is set to
zero, B is set to zero, C is set to zero, and D is set to zero. So letโs start by putting zeros
next to input A, B, C, and D. Then, we can see that inputs A and
B are the inputs to the first OR gate. So we can work out what the output
of that OR gate is going to be. In other words, what E is going to
be using the truth table for an OR gate.

We can see that when both inputs to
an OR gate are set to zero, then the output is set to zero as well. And therefore, we can see that the
output of the first OR gate, which is E, is going to be zero. And then, we can do the same thing
for inputs C and D. These are the inputs to the second
OR gate. And since both of those are set to
zero, we can see that the output of the OR gate, which is F, is going to be zero as
well. But then what we have going on is
that E and F become the inputs to the AND gate. And since in this particular case
both of them are set to zero, we can see that when both inputs to an AND gate are
set to zero, the output of the AND gate is zero as well. And hence we can say that the
output of the entire logic circuit itself is set to zero. But then, that output value
corresponds to ๐. And so at this point, we figured
out the answer to the first question. The value of ๐ in the table is
zero. And so having answered this, we can
move on to the next question.

What is the value of ๐ in the
table?

Now we can see that ๐ is the
output value that corresponds to when input A is zero, input B is zero, input C is
zero, and input D is one. So letโs put the values zero, zero,
zero, and one next to inputs A, B, C, and D, respectively. Then we can see that the inputs to
the first OR gate are set to zero. So looking at our OR gateโs truth
table, we see that when both inputs are set to zero, the output is zero as well. In other words, in this particular
case, the value of E is zero. And then, we can see that inputs C
and D are set to zero and one, respectively. And so we can see that when the
first input is set to zero but the second input is set to one for an OR gate, the
output of the OR gate is one. And so in this case, the value of F
is one. And so now what we have going on is
the value of E is zero. And the value of F is one. And those are becoming the inputs
to the AND gate. And so when the first input to the
AND gate is set to zero and the second input is set to one, the output of the AND
gate is zero. And so at this point, we found the
value of ๐ in our table. The value of ๐ is zero.

Moving onto the next question then,
this one asks us what is the value of ๐ in the table?

Now, ๐ corresponds to when input A
is zero, input B is zero, input C is one, and input D is one. And so if we set those values next
to inputs A, B, C, and D, respectively, we can see that an input of two zeros leads
an output of zero for an OR gate. So the value of E is zero whereas a
set of two ones as the inputs to an OR gate leads to an output of one. And so the value of F is one. But then, weโve already seen that
if the first input to an AND gate is zero and the second is set to one, then the
output of the AND gate is going to be zero. And so our value of ๐ is zero. So we found the answer to this
question.

Moving on to our final question
then, what is the value of ๐ in the table?

Now we can see that ๐ corresponds
to an input set of zero, one, one, and zero. So writing these down next to
inputs A, B, C, and D, we see that when the first input is zero and the second input
is one for an OR gate, the output is one. And so we see that the value of E
is one. Secondly, we can see for the lower
OR gate if the first input is one and the second input is zero, then the output is
one as well. So our value of F is one. And so now whatโs happening is that
the two inputs to the AND gate, E and F, are both set to one. And we can see that when both
inputs are set to one for an AND gate, the output is one as well. Therefore, at this point, we found
the value of ๐ in our table. So now that weโve calculated all
the values necessary, we can get rid of the columns for E and F. And we can recall that ๐ was zero,
๐ was zero, ๐ was zero, and ๐ was one.