Video Transcript
The diagram shows a logic circuit
consisting of three AND gates. How many of the inputs must have a
value of zero in order for the output to have a value of zero?
Okay, so we′ve got this logic
circuit here which is made up of three AND gates. The circuit has four inputs labeled
as A, B, C, and D. And we′re being asked to work out
how many of these inputs must have a value of zero in order for this output to have
a value of zero. We can label the output value of
zero on our diagram. And then in order to work out how
many of these four inputs must have a value of zero in order to get this output of
zero, we need to recall how an AND gate works.
The output of an AND gate is one
only if both of the two inputs to that AND gate have a value of one; otherwise, the
output of the AND gate is zero. So that means that the output of an
AND gate is zero if either or both of the inputs to it have a value of zero. If we look at the logic circuit
we′ve been given, we can see that inputs A and B are the two inputs to this top AND
gate, while input C and D are the two inputs to this bottom AND gate here. Then, the output from each of these
first two AND gates becomes one of the inputs for this third AND gate over here. And the output from this third AND
gate is the overall output of the logic circuit, which we′re told must have a value
of zero.
Since we know that the output of an
AND gate can only have a value of one if both of the two inputs are one, this means
that in order for this third AND gate to have an output value of zero, at least one
of its two inputs must be zero. Because if both of the inputs had a
value of one, then the output of the AND gate would be one. We′re being asked to work out how
many of the inputs to the circuit must have a value of zero. In other words, that′s the minimum
number of these inputs that must be zero in order to get this output of zero. The minimum number of inputs to
this third AND gate that must have a value of zero in order to get an output of zero
is one since the output of an AND gate will be zero in all cases, except the one
where both inputs have a value of one.
So, it only takes one of the inputs
being zero in order to ensure that the output of the AND gate will be zero. Both these two inputs to this third
AND gate are completely equivalent. We can say that these inputs in the
logic circuit have a kind of symmetry because each input value is created in exactly
the same way as an output of another AND gate with two inputs of its own. In other words, the part of the
circuit leading up to this upper input looks exactly the same as the part of the
circuit leading up to this lower one. What all of this means is that it
makes no difference at all which of these two inputs we choose to have a value of
zero.
Let′s suppose that the upper input
is one and the lower input is zero. Since the upper input to the third
AND gate is the output of this top AND gate here, then the output of this top AND
gate must have a value of one. And similarly, since this lower
input to the third AND gate is the output of this bottom AND gate here, then the
output of the bottom AND gate must be zero. Since we know that an AND gate can
only have an output of one if both of its two inputs are one, then in order for the
output of this top AND gate to be one, input A must be one and input B must also be
one. In order for this bottom AND gate
to have an output of zero, at least one of its two inputs must be zero.
The minimum number of inputs
required for an output of zero is one. So as a minimum, either input C
must be zero or input D must be zero. It doesn′t matter which of the two
inputs has which value since we’re simply asked how many of the inputs must have a
value of zero. So, let′s suppose, for example,
that input C is one and input D is zero. Let′s now double-check everything
we′ve done by looking at this logic circuit and following the values through from
left to right.
If we start with this top AND gate,
we can see that both of its two inputs are one, which means that it must have an
output value of one, like we′ve got. The lower AND gate has one input
with a value of one and one with a value of zero. Since the output of an AND gate can
only be one if both of its inputs are one, then this lower AND gate must have an
output of zero, which again is what we′ve got. These outputs from the first two
AND gates then become the two inputs to the third AND gate in the circuit. So, this third AND gate has one
input with a value of one and one input with a value of zero. Since it doesn′t have both inputs
equal to one, then the output must be zero, which is what we were hoping for.
So, we found that it only takes one
of the four inputs to have a value of zero in order for the output to the circuit to
have a value of zero. We can also easily check that it
doesn′t matter which of these four inputs is the one to have a value of zero. For example, let′s suppose that
input A is zero while inputs B, C, and D are all equal to one. In this case, the top AND gate has
one input of zero and one input of one. And so, its output will have a
value of zero. Meanwhile, both of the inputs to
this bottom AND gate are now one, which means that its output will have a value of
one. In this case, the third AND gate’s
upper input is zero and its lower input is one. So, that′s one input of zero and
one input of one, which means an output of zero.
Whichever of these four inputs we
choose to have a value of zero, the output of the logic circuit will be zero. If either input A or input B is
zero, then this top AND gate will have an output of zero, while if either input C or
D is zero, then the bottom AND gate will have an output of zero. In either case, at least one output
is zero, which means that at least one of these two inputs to the third AND gate is
zero. And so, the output of this AND gate
and the output to the circuit will have a value of zero.
So then, our answer to this
question is that in order for the circuit to have an output value of zero, we only
need one of the inputs to the circuit to have a value of zero.