Boolean Algebra II

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Last time we described the 7 laws of Boolean Algebra; 5 of which can help in minimizing functions. We also proved 2 add’l laws that were useful; X+1=1, X+X=X Notice the similarity between the two entries under each Axiom. Commutative a + b = b + a ab = ba Associative ( a + b ) + c = a + ( b + c ) ( ab ) c = a ( bc ) Distributive a + ( bc ) = ( a + b )( a + c ) a ( b + c ) = ( ab ) + ( ac ) Identity a + 0 = a a* 1 = a Complement a + a = 1 a * a = 0 Or with 1 And with 0 a + 1 = 1 a *0= 0 Idempotent a + a = a a * a = a Boolean Algebra II
Duality Theorem: The Dual of every theorem is a valid theorem. The dual is constructed by interchanging: 1 0 + * Or With 1: X+1=1 Dual of Or With 1:X*0=0 (And With 0) Idempotent: X+X=X Dual of Idempotent:X*X=X One last useful Theorem: DeMorgan’s Law Boolean Algebra II

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DeMorgan’s Law b a b a + = Dual of DeMorgan’s Law b a b a = + a b a b a b a b Gate Equivalency Boolean Algebra II
a b a b a b a b Gate Equivalency Q:Why is this useful? A:It allows us to build functions using only one gate type. Q:Why do we use NAND/NOR gates to build functions rather than AND/OR logic? A:NAND/NOR gates are physically smaller and faster. Q:How does Gate Equivalency help when drawing schematics with one gate type? A:Let’s look at three approaches. Boolean Algebra II

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One approach is to construct AND/OR logic and replace each gate with it’s equivalent using the figures shown below. X+Y X·Y X Y X Y X•Y X·Y X+Y X Y X Y X+Y X Y X Y X Y X Y Function Gate NAND/Inverter Equivalent X•Y X+Y X•Y X+Y X+Y X•Y X+Y = X•Y X•Y = X•Y X+Y = X•Y X•Y = X•Y Boolean Algebra II
The drawback to such an approach is that you may use more than the minimal number of gates.

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