Video Transcript
The diagram shows a logic circuit
consisting of three OR gates. A row of the truth table for this
circuit is shown below, indicating the output value for one of the possible
combinations of input values. What is the value of 𝑟 in the
table?
The question is asking us to work
out the value of one of the inputs in this combination of three OR gates. Before we start to tackle this
question, let’s remind ourselves how a single OR gate works.
Recall that an OR gate is a kind of
binary logic gate. It’s called a binary gate because
the inputs and output can each take one of two values, zero or one. An OR gate has two inputs and one
output. It’s called an OR gate because it
outputs a value of one if either this input or this input has a value of one. The gate also outputs a value of
one if both inputs are one. So, in all these three cases, the
output of the OR gate will be one. This gate will only output a value
of zero if both of the inputs are zero.
In this question, we’ve been given
a combination of three OR gates. The OR gates are connected so that
the inputs to the third gate are the outputs of the first two gates. Let’s start by labeling the diagram
with the information given to us in the table. We know that inputs A, B, and D
each have a value of zero, and we also know that the final output has a value of
one. Input C has a currently unknown
value labeled as 𝑟. To answer this question, we need to
work out whether 𝑟 is equal to zero or one.
Let’s start by looking at this gate
here, since we know the values of both of its inputs. Both of these inputs are zero. So, what does that tell us about
the output of this gate? Well, recall that an OR gate
outputs a one if either or both of its inputs are one and outputs are zero if both
inputs are zero. Since both of these inputs are
zero, this OR gate will give an output of zero.
Next, let’s look at this gate
here. This upper input is equal to this
output from the gate we just considered. So, we know this input has a value
of zero. We also know that the output of
this gate has a value of one. We can use this information to work
out the value of this lower input here. Recall that an OR gate only outputs
a value of one if either or both of the inputs are one. Since this upper input is zero,
then in order for the OR gate to output a value of one, this lower input must be
equal to one.
Finally, let’s look at this gate
here. The output of this gate is equal to
the lower input of this right-hand gate, which we’ve just worked out. So, this output has a value of
one. We know that this input, input D,
has a value of zero and this input, input C, has an unknown value 𝑟. This is exactly like we had with
the OR gate we considered previously.
Since this lower input is zero and
the output is one, we know that this upper input here must have a value of one. If instead this input were zero,
then the output of the gate would be zero and not one. This input is the value of 𝑟,
which is what we were asked to find. Our answer then is that 𝑟 has a
value of one.
