### Video Transcript

The diagram shows two OR gates
connected as part of a logic circuit. The truth table shows the output
for the various combinations of inputs. What is the value of ๐ in the
table? What is the value of ๐ in the
table? What is the value of ๐ in the
table? What is the value of ๐ in the
table?

Okay, so weโve got four parts to
this question, and each of these parts is asking us to find the value of one of the
quantities ๐, ๐, ๐, and ๐ in this truth table. Weโll tackle these parts one at a
time, starting with this first one. This says, what is the value of ๐
in the table? If we look at our truth table, we
can see that ๐ is one of the possible values for the input ๐ถ in this logic
circuit. Specifically, the second row of the
truth table tells us that ๐ is the value of input ๐ถ that when input ๐ด has a value
of zero and input ๐ต also has a value of zero, then the combination of these three
inputs means that the logic circuit as a whole has an output of one.

Knowing that weโre considering the
second row in the table, letโs see how this applies to our logic circuit. We can see that the circuit
consists of two OR gates. Weโve got this left-hand OR gate
whose inputs are input ๐ต and input ๐ถ. The output from this first OR gate
then becomes one of the two inputs for the right-hand OR gate. The other input to this second OR
gate is input ๐ด. And then its output is the overall
output from this combination of OR gates. So thatโs the final column of our
truth table.

Letโs put the values from the
second row of the truth table onto our diagram. We have that input ๐ด is equal to
zero, input ๐ต is equal to zero, input ๐ถ has a value of ๐, which is what weโre
trying to find, and finally the output has a value of one.

To understand whatโs going on here,
we need to recall how an OR gate works. An OR gate is a type of logic gate
that gives an output of one if either of the inputs or both of them have a value of
one. Otherwise, so if both of the inputs
have a value of zero, then the output is zero. Just as we have this truth table
for the whole of the circuit that weโre shown, we can also write out a truth table
for an individual OR gate. An OR gate has two inputs, which
weโve generically labeled here as a first input and a second input. And the OR gate gives a single
output value dependent on these two inputs according to this logic weโve explained
here.

If the first input has a value of
zero and the second input is also zero, then this first bullet point doesnโt apply
because neither input has a value of one. This means that we must be looking
at the second bullet point, and so in this case the output of the OR gate is
zero. If the first input is equal to zero
but the second input is equal to one, then now one of our two inputs does have a
value of one, and so the OR gate has an output of one. Similarly, if the first input is
one and the second input is zero, then thatโs at least one input with a value of
one. And so weโve got an output of
one. The last case to consider is that
the first input is one and the second input is also one. This first bullet point tells us
that the output of an OR gate is one if either or both of the inputs are one. In this case, weโve got both inputs
equal to one. And so the output must be one.

With this table in mind, letโs now
have a look at the diagram. Specifically, weโre going to start
by considering this right-hand OR gate. We know that it has an output value
of one. And we also know that one of its
inputs, input ๐ด, has a value of zero. We know from our OR gate truth
table that if both inputs have a value of zero, then the output must be zero. Now thatโs not the case here
because our output value is one. So since input ๐ด is zero, this
means that the other input to this right-hand OR gate must have a value of one. This lower input to the right-hand
OR gate comes from the output of the left-hand OR gate. So we know that the output of this
left-hand OR gate is equal to one.

If we now consider the inputs to
the left-hand OR gate, we can use exactly the same logic as we used for the
right-hand gate. Since the output has a value of
one, then our truth table tells us that it canโt be the case that both inputs have a
value of zero. Since we know that input ๐ต is
equal to zero, then this means that input ๐ถ cannot be equal to zero. And so ๐, which is the value of
input ๐ถ in this case, must be equal to one.

Okay, now letโs move on to the
second part of the question. What is the value of ๐ in the
table?

Okay, so now weโre thinking about
the quantity ๐, which is one of the possible output values for this combination of
gates. The fourth row of this table tells
us that ๐ is the output value we get when input ๐ด is zero, input ๐ต is one, and
input ๐ถ is one.

Letโs put these values on our
diagram. If we look at this left-hand OR
gate, we can see that input ๐ต is one and input ๐ถ is one. And so weโre looking at the bottom
row of our truth table for a single OR gate. We can see that both inputs equal
to one mean weโve got an output of one. And we know that this output then
becomes the lower input for the right-hand OR gate.

If we now look at this right-hand
gate, we can see that the upper input, input ๐ด, is equal to zero, while the lower
input is equal to one. The middle two rows in our OR gate
truth table tell us that so long as at least one input has a value of one, the
output of an OR gate will have a value of one. ๐ is the output value that we get
when weโve got one input of zero and one input of one. And so ๐ must be equal to one.

Now letโs look at the third part of
the question. What is the value of ๐ in the
table?

If we look at the truth table, we
can see that ๐ is one of the possible values of input ๐ด. In particular, itโs the value of
input ๐ด that when input ๐ต is zero and input ๐ถ is also zero, the circuit as a
whole gives an output value of one. So letโs add the values from this
row of the truth table to our diagram. If we start by looking at the
left-hand OR gate, we can see that both of its inputs have a value of zero. This means that the output from
this OR gate must be zero. And therefore, the lower input to
this right-hand OR gate must be zero. We can see that the output of this
right-hand OR gate is equal to one. And we know that its lower input is
zero.

From our OR gate truth table, we
can see that if both inputs were zero, then the output would be zero. So this means that the other input,
input ๐ด or our value of ๐, cannot be zero. So then we must have that ๐ is
equal to one.

Letโs now move on to the fourth and
final part of the question. What is the value of ๐ in the
table?

Looking at the truth table weโve
been given, we can see that ๐ is the output value when input ๐ด is one, input ๐ต is
one, and input ๐ถ is zero. Letโs go ahead and put the values
from this row of the table onto our diagram. Weโll begin by looking at the
left-hand OR gate. Itโs got one input, input ๐ต, equal
to one, and the other input, input ๐ถ, equal to zero. Since at least one input is one,
then the output of this OR gate is equal to one. And so the lower input to the
right-hand OR gate is also one. If we now look at the right-hand OR
gate, we can see that both of its inputs are equal to one, and so its output must be
equal to one. The output of this right-hand OR
gate is our value of ๐ , and so we have found that ๐ is equal to one.