Video Transcript
The diagram shows a logic circuit
consisting of three AND gates. One row of a truth table 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?
This question is asking us to work
out the value of input C in this combination of three AND gates. Before we start to tackle this
question, let’s remind ourselves how a single AND gate works.
Recall that an AND gate is a kind
of binary logic gate. It’s called a binary gate because
the inputs and output can take one of two values, zero or one. An AND gate has two inputs and one
output. It’s called an AND gate because it
only outputs a value of one if both this input and this input have a value of
one. If either or both of the inputs
have a value of zero, then the output of the gate is zero.
In this question, we’ve been given
a combination of three AND gates. The AND gates are connected so that
the inputs to the third gate are the outputs of the first two gates. Let’s begin by labeling the diagram
with the information given to us in the table. We know that inputs A, B, and D all
have a value of one and that the final output has a value of zero. Input C has a currently unknown
value of 𝑟. To answer this question, we need to
work out whether 𝑟 has a value of zero or one.
Let’s start by looking at this AND
gate here with inputs A and B, since we know the values of both of these inputs. Both these two inputs are equal to
one, which means that the output of this gate will also be one. Then, the output of this AND gate
is connected to the upper input of the final AND gate, so we know that this input
has a value of one.
So, what does that tell us about
this input here? Well, let’s recall that an AND gate
only outputs a value of one if both of its inputs are one. If both this upper input and this
lower input were equal to one, then the gate would output a value of one. Since this upper input is one, but
the gate still outputs a value of zero, we know that this lower input must have a
value of zero. This input is equal to the output
of this AND gate here, with inputs C and D.
If we look at this gate, we see
that we have one input with a value of one and an output with a value of zero. We need to work out the value, 𝑟,
of input C. We can apply the exact same logic
that we used with the previous gate. Since this lower input is a one,
but the gate still outputs a zero, we know that 𝑟 must have a value of zero. If the input were one, then the
gate would instead output a one. So we have worked out that 𝑟 must
have a value of zero. This is the final answer to this
question.