The Laws of Boolean

The Laws of Boolean

As well as the logic symbols "1" and "0" being used to represent a digital input or output, we can also use them as constants for a permanently "Open" or "Closed" circuit or contact. Laws or rules for Boolean Algebra expressions have been invented to help reduce the number of logic gates needed to perform a particular logic operation resulting in a list of functions or theorems known commonly as the Laws of Boolean. Examples of these individual rules or laws are given in the following table.

Truth Tables for the Laws of Boolean

Expression	Description	Equivalent Circuit	Law or Rule
A + 1 = 1	A in parallel with closed = CLOSED		Annulment
A + 0 = A	A in parallel with open = A		Identity
A . 1 = A	A in series with closed = A		Identity
A . 0 = 0	A in series with open = OPEN		Annulment
A + A = A	A in parallel with A = A		Indempotent
A . A = A	A in series with A = A		Indempotent
NOT A = A	NOT NOT A (double negative) = A		Double Negation
A + A = 1	A in parallel with not A = CLOSED		Complement
A . A = 0	A in series with not A = OPEN		Complement
A+B = B+A	A in parallel with B = B in parallel with A		Commutative
A.B = B.A	A in series with B = B in series with A		Commutative
A+B = A.B	invert and replace OR with AND		de Morgan's Theorem
A.B = A+B	invert and replace AND with OR		de Morgan's Theorem

Description of the Laws and Theorems

• Annulment Law - A term AND´ed with a "0" equals Zero and a term OR´ed with a "1" will equal One.
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• Identity Law - A term OR´ed with a "0" or AND´ed with a "1" will always equal that term for example, A+0 = 0, A.1 = 1
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• Indempotent Law - An input AND´ed with itself or OR´ed with itself is equal to that input for example, A.A = A, A+A = A
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• Complement Law - A term AND´ed with its complement equals "0" and a term OR´ed with its complement equals "1" for example A. A = 0, A+A = 1
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• Commutative Law - The order of application of two separate terms is not important for example, A.B = B.A, A+B = B+A
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• Double Negation Law - A term that is inverted twice is equal to the original term.
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• de Morgan´s Theorem - There are two "de Morgan´s" rules or theorems,
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• (1) Two separate terms NOR´ed together is the same as the two terms inverted (Complement) and AND´ed for example, A+B = A. B.
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• (2) Two separate terms NAND´ed together is the same as the two terms inverted (Complement) and OR´ed for example, A.B = A +B.

Boolean Algebra Functions

Using the information above, simple 2-input AND, OR and NOT Gates can be represented by 16 possible functions as shown in the following table.