Unformatted text preview: Postulate 9 A + A = A (not 2A) mmmwio A+A=1
Postulate 11 A + 1 = 1
Postulate 12 A + 0 = A For Postulate 9, it can be seen that regardless of
whether element A is 0 or 1, the result will be
identical because either element is the same. For
Postulate 10, the result will alwals be a true
statement (1) because if A is true, A is false and
viceversa. Postulate 11 states that the result will
always be 1 regardless of whether A is 1 or 0
because the one element is true. Postulate 12 says
that the result will be determined by the value of
A; if A is true, the result is true, and if A is false
the result is false. The OR operation is performed using the logic
OR gate shown in Fig. A3 (a). The truth table of
(b) shows the fundamental relationships of true
and false conditions applied to the gate. The OR
operation pertains to any number of variables.
The general statement of the OR function is: A+B+C+....=X INPUTS { X=A+B
[I] 1 ‘l
'I D
0 II
D D Fig. A3 Combined Operations The basic NOT, AND and OR operations can be
combined to give four additional operations that
are unique to Boolean algebra. These are: NAND,
NOR, EXCLUSIVE OR and EXCLUSIVE NOR,
which are described in the subsequent paragraphs. The NAND Function. The NAND operation is the
complement of the AND operation. The term
NAND is a contraction of NOT AND. The NAND
operation is indicated by a NOT symbol placed
over a logic expression that results from AND and
NOT operations. For example, the NAND logic
expression for variables A and B is A B: X.
It can also be stated that X— — A B. The result
indicates that the variables are ﬁrst combined into
a single term or combination of terms, which is
then complemented. The four possible combinations of the NAND
operation can be stated in terms of the elements 1
and 0 used as two independent variables. 1AND1=T=0
1AND0=6=1 Appendix A A—3 * OAND1=6=1
0AND0=6=1 It can be seen from these results that the NAND
function is the inverse of the AND function. For
the NAND function to be true, one or more of the
variables must be false. The function is false only
when all the variables are true. The logic circuit for the NAND function is an
AND gate followed by an inverter as shown in
Fig. A4 (a). These two circuits are usually combined
into a single N AND gate as shown by the symbol of
(b). The small circle at the output of the NAND
gate symbol indicates inversion. The truth table of
(c) shows the fundamental true and false relation
ships of the NAND gate. The general forms of the
NAND equation are: ABCum=X
OI
X=ABCmu A
INPUTS 5 w “3 >. (Hi (C) Fig. A4 The NOR Function. The NOR operation is the
complement of the OR operation. The term NOR
is a contraction of NOT OR. The NOR operation is
indicated by a NOT symbol placed over a logic
expression that results from OR and NOT opera—
tions. For example, the NOR log_ic expression for
variables A and B is A + B = X. It can also be
stated that X = A + B. The result indicates that a
single term or combination of terms is first
obtained for the variables, which is then com
plemented. The truefalse elements 1 and 0, used as two
independent variables, provide the four fundamental
logic relationships of the NOR function. 10R1=T=0
10R0=i=0
00R1=i=0
00R0=6=1 These relationships clearly show that the NOR
operation is the inverse of the OR operation. For ...
View
Full Document
 Fall '05
 Myer,B

Click to edit the document details