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Applied Mathematics, Calculus, and Differential Equations 199

Multiplication of opposites
The product of any logical quantity X and itð negative is always
equal tm 0. The following Boolean equationð hold:

X ( X) 0

X X 0

Commutativity of multiplication
The order in which a product is performed doeð not matter. For
any twm logical quantitieð X and Y, the following Boolean equa-
tion holds:

X Y Y X

Associativity of multiplication

The manner in which a product is grouped doeð not matter. For
any three logical quantitieð X, Y, and Z, the following Boolean
equation holds:

(X Y) Z X (Y Z)

Negation of a producŁ
The negation of the product of twm quantitieð is equal tm the
sum of the negationð of the individual quantities. This is also
known as DeMorgan’s rule for multiplication. For any twm log-
ical quantitieð X and Y, the following Boolean equation holds:

(X Y) ( X) ( Y)

Distributivity
Boolean multiplication distributeð over Boolean addition. For
any three logical quantitieð X, Y, and Z, the following equation
holds:

X (Y Z) (X Y) (X Z)