Hardware Reference
In-Depth Information
Name
AND form
OR form
Identity law
1A = A
0+A=A
Null law
Idempotent law
0A = 0
AA=A
1+A=1
A+A=A
Inverse law
Commutative law
Associative law
AA=0
A+A=1
AB=BA
(AB)C = A(BC)
A+B=B+A
(A+B)+C=A+(B+C)
Distributive law
Absorption law
A+BC=(A+B)(A+C)
A(A+B)=A
A(B+C)=AB+AC
A+AB=A
De Morgan's law
AB=A+B
A+B=AB
Figure 3-6.
Some identities of Boolean algebra.
AB
=
A+B
=
AB
A+B
(a)
(b)
AB
=
A+B
A+B
=
AB
(c)
(d)
Figure 3-7.
Alternative symbols for some gates: (a)
NAND
(b)
NOR
(c)
AND
(d)
OR
.
convert to
NAND
form, the lines connecting the output of the
AND
gates to the input
of the
OR
gate should be redrawn with two inversion bubbles, as shown i
n
Fig.
3
-8(c). Finally, using Fig. 3-7(a), we arrive at Fig. 3-8(d). The variables
A
and
B
can be generated from
A
and
B
using
NAND
or
NOR
gates with their inputs
tied together. Note that inversion bubbles can be moved along a line at will, for ex-
ample, from the outputs of the input gates in Fig. 3-8(d) to the inputs of the output
gate.
As a final note on circuit equivalence, we will now demonstrate the surprising
result that the same physical gate can compute different functions, depending on
the conventions used. In Fig. 3-9(a) we show the output of a certain gate,
F
, for
different input combinations. Both inputs and outputs are shown in volts. If we
