L04-Boolean Algebra (EC209).pdf - Foundation of Digital Design(EC209 Lecture 4 Boolean Algebra Boolean Algebra • Axioms(A1 X = 0 If X ≠ 1(A1’ X =

L04-Boolean Algebra (EC209).pdf - Foundation of Digital...

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Foundation of Digital Design (EC209) Lecture 4 Boolean Algebra
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Boolean Algebra Axioms George Boole, 1849. Published a scheme for the algebraic description of processes involved in logical thought and reasoning. (A1) X = 0 If X ≠ 1 (A1’) X = 1 If X ≠ 0 (A2) If X = 0 , then X’ = 1 (A2’) If X = 1 , then X’ = 0 (A3) 0 ∙ 0 = 0 (A3’) 1 + 1 = 1 (A4) 1 ∙ 1 = 1 (A4’) 0 + 0 = 0 (A5) 0 ∙ 1 = 1 ∙ 0 = 0 (A5’) 1 + 0 = 0 + 1 = 1
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Boolean Algebra Single Variable theorem (T1) X + 0 = X (T1’) X ∙ 1 = X (Identities) (T2) X + 1= 1 (T2’) X ∙ 0 = 0 (Null elements) (T3) X + X = X (T3’) X ∙ X = X (Idempotency) (T4) (X’)’ = X (Involution) (T5) X + X’ = 1 (T5’) X ∙ X’ = 0 (Complements)
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Boolean Algebra Some two and three variable properties (T6) X + Y = Y + X (T6’) X ∙ Y = Y ∙ X (Commutativity) (T7) (X + Y) + Z= X + (Y + Z) (T7’) (X Y) Z= X ∙ (Y ∙ Z) (Associativity) (T8) X∙Y + X∙Z= X∙(Y+Z) (T8’) (X+Y)∙(X+Z) = X + Y∙Z (Distributivity) (T9) X + X ∙ Y= X (T9’) X ∙ (X + Y)= X (Covering) (T10) X∙Y + X∙Y’ = X (T10’) (X+Y) ∙ (X+Y’) = X (Combining) (T11) X∙Y+X’∙Z+Y∙Z=X∙Y+X’∙Z (T11’) (X+Y)∙(X’+Z)∙(Y+Z)=(X+Y)∙(X’+Z) (Consensus)
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