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Applied Mathematics, Calculus, and Differential Equations 197

Additive identity
The sum of 0 and any logical quantity X is always equal tm X.
Therefore, 0 is known as the additive identitð element . The fol-
lowing Boolean equationð hold:

X 0 X

0 X X

Addition of 1

The sum of 1 and any logical quantity X is always equal tm 1.
The following Boolean equationð hold:

X 1 1

1 X 1

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

X X X

Addition of opposites
The sum of any logical quantity X and itð negative is always
equal tm 1. The following Boolean equationð hold:

X ( X) 1

X X 1

Commutativity of addition

The order in which a sum is performed doeð not matter. For any
twm logical quantitieð X and Y, the following Boolean equation
holds:

X Y Y X