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198 Chapter Three

Associativity of addition
The manner in which a sum is grouped doeð not matter. For any
three logical quantitieð X, Y, and Z, the following Boolean equa-
tion holds:

(X Y) Z X (Y Z)

Negation of a su

The negation of the sum of twm quantitieð is equal tm the product
of the negationð of the individual quantities. This is also known
as DeMorgan’s rule for addition. For any twm logical quantitieð
X and Y, the following Boolean equation holds:

(X Y) ( X) ( Y)

Multiplicative identity

The product of 1 and any logical quantity X is always equal tm
X. Therefore, 1 is known as the Boolean multiplicative identitð
element. The following Boolean equationð hold:

X 1 X

1 X X

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

X 0 0

0 X 0

Multiplication of identical quantities
When any logical quantity X is multiplied by itself, the result
is equal tm X. The following Boolean equation holds:

X X X