Electronics guide > Digital integrated circuits I > The Boolean Way — It’s Logical!
The Boolean Way — It’s Logical!Now we already know that the Boolean statement for a NAND gate is:
But we can also derive another Boolean statement from the above circuit, by the
very fact that the inputs to the OR gate in Figure 10.42 are inverted before they
reach the OR gate, such that the final output Z is equal to logic 1 when NOT A OR
NOT B OR NOT C are logic 1.
So, the Boolean statement:
also expresses the NAND gate.
In other words:
NOR from AND and NOT
And, just for the sake of completion, we can also work out that a NOR gate can
be created by inverting the inputs of an AND gate.
As we’ve done so far in this book, we’ll do this first experimentally, building
up the circuit and checking results. Then we’ll do it mathematically.
Figure 10.43 shows a circuit we can use to do this. It’s basically a two input
AND gate, the inputs of which have been inverted by NOT gates.
Figure 10.44 shows a possible breadboard layout you can follow to build the experimental
circuit on, while Figure 10.45 is an uncompleted truth table for you to complete
with your results, and Figure 10.46 is the complete truth table whose results should
match yours.
Figure 10.43 The experimental circuit to prove that a NOR gate
can be created from an AND gate whose inputs are inverted first
Figure 10.44 A breadboard layout for the circuit in Figure
10.43
Figure 10.45 An incomplete truth table for you to record your
results
Figure 10.46 A completed truth table for the circuit in Figure
10.43 — showing how we have made a NOR gate from an AND gate with NOT gates at its
inputs
And, for the sake of completeness, Figure 10.47 shows a three input AND gate
with NOT gates at its inputs, and the circuit’s truth table, to show that we can
make a NOR gate of any number if inputs from an AND gate of that number of inputs
together with the requisite number of inverters.
On the other hand, of course, and even better than this, by the same means of
deduction we used for other circuits combining logic gates, we can calculate that
the standard NOR Boolean statement:
is also the same as another Boolean statement:
Figure 10.47 A three input AND gate converted to a NOR gate
with NOT gates
In other words:
And — there — we have just proved mathematically that you can make a NOR gate
from AND and NOT gates.
So, there’s no doubt about it, all digital electronic logic gates can be created
from a number of other digital electronic logic gates.
What’s more, it’s not beyond the realms of possibility to work out for ourselves
that these very logic gates, that can make any other logic gates, can be combined,
and combined again, to make up ever more complex circuits. And, indeed they are
— as we’ll see in the next chapter! For now, have a go at the quiz over the page,
to see if you’ve been taking notice…
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