Digital Systems - Laws Of Boolean Algebra

de Morgan's Theorem - There are two "de Morgan's" rules or theorems

(1) Two separate terms NOR'ed together is the same as the two terms inverted (Complement) and AND'ed (2) Two separate terms NAND'ed together is the same as the two terms inverted (Complement) and OR'ed

Boolean Postulates - While not Boolean Laws in their own right, these are a set of Mathematical Laws which can be used in the simplification of Boolean Expressions.

0 . 0 = 0 A 0 AND'ed with itself is always equal to 0 1 . 1 = 1 A 1 AND'ed with itself is always equal to 1 1 . 0 = 0 A 1 AND'ed with a 0 is equal to 0 0 + 0 = 0 A 0 OR'ed with itself is always equal to 0 1 + 1 = 1 A 1 OR'ed with itself is always equal to 1 1 + 0 = 1 A 1 OR'ed with a 0 is equal to 1

Absorptive Law - This law enables a reduction in a complicated expression to a simpler one by absorbing like terms. A + (A.B) = (A.1) + (A.B) = A(1 + B) = ? (OR Absorption Law) A(A + B) = (A + 0).(A + B) = A + (0.B) = ? (AND Absorption Law)

A + (A.B) = (A.1) + (A.B) = A(1 + B) = A (OR Absorption Law) A(A + B) = (A + 0).(A + B) = A + (0.B) = A (AND Absorption Law)

Associative Law - This law allows the removal of brackets from an expression and regrouping of the variables. A + (B + C) = (A + B) + C = ? (OR Associate Law) A(B.C) = (A.B)C = ? (AND Associate Law)

A + (B + C) = (A + B) + C = A + B + C (OR Associate Law) A(B.C) = (A.B)C = A . B . C (AND Associate Law)

Identity Law - A term OR'ed with a "0" or AND'ed with a "1" will always equal that term A + 0 = A . 1 =

A + 0 = A... A variable OR'ed with 0 is always equal to the variable A . 1 = A ...A variable AND'ed with 1 is always equal to the variable

Idempotent Law - An input that is AND'ed or OR´ed with itself is equal to that input A . A= A + A=

A + A = A... A variable OR'ed with itself is always equal to the variable A . A = A... A variable AND'ed with itself is always equal to the variable

Annulment Law - A term AND'ed with a "0" equals ___ or OR'ed with a "1" will equal ___.

A . 0 = 0 A variable AND'ed with 0 is always equal to 0 A + 1 = 1 A variable OR'ed with 1 is always equal to 1

Complement Law - A term AND'ed with its complement equals "0" and a term OR´ed with its complement equals "1" A . A' = A + A' =

A . A' = 0 A variable AND'ed with its complement is always equal to 0 A + A' = 1 A variable OR'ed with its complement is always equal to 1

Commutative Law - The order of application of two separate terms is not important A + B= A . B=

A . B = B . A The order in which two variables are AND'ed makes no difference A + B = B + A The order in which two variables are OR'ed makes no difference

Distributive Law - This law permits the multiplying or factoring out of an expression. A(B + C) = A + (B.C) =

A(B + C) = A.B + A.C (OR Distributive Law) A + (B.C) = (A + B).(A + C) (AND Distributive Law)