1 - Boolean Algebra, Minterm & Maxterm Expansion, K-Maps

1 - Boolean Algebra, Minterm & Maxterm Expansion,...

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EEN 304 Logic Design Dr. Kabuka
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EEN 304: Logic Design 2 BOOLEAN ALGEBRA Basic Operations: 1) The AND operation " ". Figure 1 F=1 iff A, B, and C are all 1. 2) The OR operation " + ". Figure 3 F=1 iff A or B (or Both) is 1. 3) The NOT operation " ". (Complement or Inverse) Figure 4 A B B A F = 0 0 0 0 1 0 1 0 0 1 1 1 A B C C B A F = 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 1 0 0 Figure 2 1 1 1 1 A B B A F + = 0 0 0 0 1 1 1 0 1 1 1 1 A A 0 1 1 0
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EEN 304: Logic Design 3 Duality: ) 0, 1, , X ..., , X F ) 1, 0, , X ..., , F(X n 1 D n 1 + + , ( , Basic Theorems Operation with 0 and 1 1. X 0 X = + 1D. X 1 X = Proof: X 0 X + X 0 X 0 0 0 0 = + 0 0 0 0 = 1 1 0 1 = + 1 1 0 1 = The Same The Same 2. 1 1 X = + 2D. 0 0 X = Idempotent Laws: 3. X X X = + 3D. X X X = Proof: X X X + X X X 0 0 0 0 = + 0 0 0 0 = 1 1 1 1 = + 1 1 1 1 = The Same The Same Involution Law: 4. X ) X ( = Proof: X X ) X ( 0 1 0 1 0 1 The Same Laws of Complementarity: 5. 1 X X = + 5D. 0 X X = Proof: X X X X + X X X X 0 1 1 1 0 = + 0 1 0 1 0 = 1 0 1 0 1 = + 1 0 0 0 1 =
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EEN 304: Logic Design 4 Commutative Laws: 6. X Y Y X + = + 6D. X Y Y X = Associative Laws: 7. Z) (Y X Z Y) (X + + = + + 7D. Z) (Y X Z Y) (X = Z Y X + + = Z Y X = Distributive Laws: 8. Z X Y X Z) (Y X + = + 8D. Z) (X Y) X Z Y X + + = + ( Proof: Z Y X Z Y Z) Y (1 X Z Y Y X Z X X Z Y X Y Z X X X Z) (X Y Z) (X X W Y W X W Y) (X Z) (X Y) (X W Z X Let + = + + + = + + + = + + + = + + + = + = + = + + = + Simplification Theorems: 9. X Y X Y X = + 9D. X ) Y (X Y) (X = + + Proof of 9D: 1a by X 0 X Y Y X 5D by 0 Y Y Since Y Y X ) Y (X Y) (X get can we 8D, From = + = + = + = + + 10. X Y X X = + 10D. X Y) (X X = + Proof of 10D: X Y) (1 X Y X X Y X X X Y) (X X = + = + = + = + 11. Y X Y ) Y (X = + 11D. Y X Y Y X + = +
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EEN 304: Logic Design 5 Example 1: Simplify the following Boolean functions. C
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1 - Boolean Algebra, Minterm & Maxterm Expansion,...

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