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3.11 LAWS OF XOR ALGEBRA 111

of the outer bar takes place only after simplification. As a general rule, DeMorgan's laws
should be applied to a function only after it has been sufficiently reduced so as to avoid
unnecessary Boolean manipulation.

3.11 LAWS OF XOR ALGEBRA

The laws of XOR algebra share many similarities with those of conventional Boolean
algebra discussed in the previous section and can be viewed as a natural extension of the
conventional laws. Just as the AND and OR laws are deduced from their respective truth
tables, the XOR and EQV laws are deduced from their respective truth tables in Figs. 3.26c
and 3.27d and are given by Eqs. (3.16) together with their truth tables:

XOR EQV
Truth Table Truth Table
X Y X@Y X Y XOY

0 0 0 0 0
0 1 1 0 1
1
1 0 \,—, I I/ ' »
1 1 0 * XOR Laws EQV Laws * 1 1

(3.16)

Here, the dual relationship between the XOR and EQV laws is established by interchanging
the 1 's and O's while simultaneously interchanging the XOR and EQV operators, as indicated
by the double arrow.
The associative and commutative laws for EQV and XOR follow from the associative
and commutative laws for AND and OR given by Eqs. (3.10) and (3.11) by exchanging
operator symbols: O for (•) and © for (+). The distributive, absorptive, and consensus
laws of XOR algebra follow from their AND/OR counterparts in Eqs. (3.12), (3.13), and
(3.14) by replacing the appropriate (+) operator symbols with the © operator symbols,
and by replacing the appropriate (•) symbols with the O symbol, but not both in any given
expression. In similar fashion, DeMorgan's laws in XOR algebra are produced by substi-
tuting O for (•) and © for (+) in Eqs. (3.15). These laws are presented as follows in dual
form and in terms of variables X , F, and Z, which may represent single or multivariable
functions:

Associative Law { } (3.17)
\(X © Y) © Z = X © (Y © Z) = X © Y © ZJ

. ¥ \XQYOZ=XQZQY =ZQXQY = ]
Commutative Laws { } (3.18)