l08 - Combinational Logic Multiple levels of...

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Unformatted text preview: Combinational Logic Multiple levels of representation: Logic equations Truth tables Gate diagrams Switching circuits Boolean algebra: tool to manipulate logic equations An algebra on a set of two elements: { , 1 } Operations: AND, OR, complement Boolean Algebra Identities: a = 0 1 a = a aa = a aa = 0 0 + a = a 1 + a = 1 a + a = a a + a = 1 ab = ba a ( bc ) = ( ab ) c a + b = b + a a + ( b + c ) = ( a + b ) + c a ( b + c ) = ab + ac a + ( bc ) = ( a + b )( a + c ) ( a + b ) = a b ( ab ) = a + b Precedence: AND takes precedence over OR. Proving Logic Equations Example: ( a + b )( a + c ) = a + bc Algebraic proof? Proof with Truth Tables: a b c a + b a + c LHS bc RHS 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Truth Tables To Logic Equations a b c out Minterms Maxterms a b c a + b + c 1 1 a b c a + b + c 1 1 a bc a + b + c 1 1 a bc a + b + c 1 1 ab c a + b + c 1 1 1 ab c a + b + c 1 1 abc a + b + c 1 1 1 abc a + b + c Sum of Products: a b c + a bc + ab c + ab c Product of Sums: ( a + b + c )( a + b + c )( a + b + c )( a + b + c ) Universality: NAND and NOR = = = = Universal: can implement any combinational function using just NAND or just NOR gates. Minimizing Logic Equations...
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l08 - Combinational Logic Multiple levels of...

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