# tutorial3 - ERG2020A Tutorial 3 Boolean Algebra(2 Karnaugh...

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ERG2020A Tutorial 3 Boolean Algebra (2) Karnaugh Map (K-map) (1)

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Basic Boolean Operators Basic gate types - AND : output is 1 if BOTH inputs are 1 ab , a · b , a b - OR : output is 1 if EITHER inputs is 1 a + b , a b - NOT : 0 1 x , x ’, ~ x NAND NOR
What is Boolean Functions? Define B = {0, 1}, given non-negative k B B

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Boolean Functions & Truth Table Ex.1: verify F = X Y + Z X Y Z F 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1 B B
Representations for Boolean Functions X Y Z F 0 0 0 1 0 0 1 0 0 1 0 1 0 1 1 0 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1 X Y Z F F = X Y + Z

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Duality An algebraic expression is valid after DUAL operations: AND OR, 0 1 Example: x + 0 = x x ·1 = x x ·( y + z ) = xy + xz x + y · z = ( x + y )·( x + z )
Identities Boolean “Instinct” DeMorgan’s dual * “short” eats “long” * common factor

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Consensus Theorem xy + x z + yz = xy + x z proof: xy + x z + yz = xy + x z + yz ( x + x ’) = xy + x z + xyz + x yz = xy + xyz + x z + x yz = xy (1 + z ) + x z
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tutorial3 - ERG2020A Tutorial 3 Boolean Algebra(2 Karnaugh...

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