5 - ECSE-2610 Computer Components & Operations (COCO)...

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ECSE-2610 Computer Components & Operations (COCO) Fall 2006 Part 5: Boolean Algebra and Switching Theory
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Reminder of Basic Boolean Operations X (NOT X) Sometimes written X‘. X • Y (X AND Y) Sometimes written XY X + Y (X OR Y) X, X, and Y are called literals . Definition : The dual of a Boolean equation is derived by Replacing each AND operation by OR Replacing each OR operation by AND Replacing each 0 by 1 Replacing each 1 by 0 Literals are left unchanged. Law of Duality: For any equation equation that is true, its dual is also true! Example : The dual of X + 0 = X is X • 1 = X
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These results can all be proved by truth tables. Some can also be derived from the others. The right side ones are the duals of the left side ones. Identity Elements (0, 1) 1. X + 0 = X 1D. X • 1 = X Proofs by Truth Table (aka “Perfect Induction”) X 0 0 0 0 1 X + 0 0 1 X 1 1 1 0 1 X 1 0 1
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Nullification Laws 2. X + 1 = 1 2D. X • 0 = 0 Proofs by Truth Table Idempotent Laws 3. X + X = X 3D. X • X = X X 0 0 0 0 1 X + 1 1 1 X 1 1 1 0 1 X 0 0 0
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Involution Law Proof by Truth Table 4. (X')' = X Laws of Complementarity 5. X + X' = 1 5D. X • X' = 0 Proofs by Truth Table X' (X')' 0 1 1 0 X 0 1 X' X+X' 0 1 1 0 X 1 1 X' X•X' 0 1 1 0 X 0 0
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Laws & Theorems continued Commutative Laws 6. X + Y = Y + X 6D. X • Y = Y • X Associative Laws 7. (X + Y) + Z = X + (Y + Z) 7D. (X • Y) • Z = X • (Y • Z) Distributive Laws 8. X • (Y+ Z) = (X • Y) + (X • Z) 8D. X+(Y• Z) = (X+Y)•(X + Z) Covering 9. X + (X • Y) = X 9D. X • (X + Y) = X Proof of 9 by Truth Table: X • Y 0 0 1 1 X Y 0 1 0 1 0 0 0 1 X+(X • Y) 0 0 1 1
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Laws & Theorems continued Combining I 10. X • Y
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5 - ECSE-2610 Computer Components & Operations (COCO)...

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