Page 144 - Engineering Digital Design

3.11 LAWS OF XOR ALGEBRA 115

The most obvious application of Corollaries I and II is in operator interchange as demon-
strated by the following four examples:

[1] AB + BC = (Afi)0(flC),

where a = AB, fi = BC, and a • ft = 0 by Corollary I.
[2] (A + B + X) • (A + B + C + Y) = (A + B + X) Q (A + B + C + Y),

where a = (A + B + X), ft = (A + B + C + 7) and a + ft = 1 according to Corollary II.

[3] a+b®bc = a + b + bc = a + b + c,

where Corollary I has been applied followed by the absorptive law in Eqs. (3.13).

[4] (Xf) 0 (X + Y + Z) = (XY)(X + Y + Z) = XYZ

Here, Corollary II is applicable since XY = X + 7, and the result follows by using the
AND and OR laws given by Eqs. (3.9).

3.11.2 Summary of Useful Identities

The laws of XOR algebra have been presented in the foregoing subsections. There are several
identities that follow directly or indirectly from these laws. These identities are useful for
function simplification and are presented here in dual form for reference purposes.

®x = x0x = n

QX = X0X = 0|
XQX =
o x = *
oox = x

= Y = Y 1

(X + 7) O X = (X + Y) O Y = 0 0 (X + )

(X + Y)(Y + Z)(X + Z) = (X + 7) O (Y + Z) O (X + Z)

Note that in these identities, either X or Y or both may represent multivariable functions or
single variables of any polarity (i.e., either complemented or uncomplemented).