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Boolean Algebra

Boolean Algebra. 350151 – Digital Circuit 1 Choopan Rattanapoka. Basic Theorems (1). Operation with 0 and 1 X + 0 = X X + 1 = 1 X • 0 = 0 X • 1 = X Idempotent laws X + X = X X • X = X. Basic Theorems (2). Involution laws (X’)’ = X Laws of complementarity X + X’ = 1

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Boolean Algebra

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  1. Boolean Algebra 350151 – Digital Circuit 1 ChoopanRattanapoka

  2. Basic Theorems (1) • Operation with 0 and 1 • X + 0 = X • X + 1 = 1 • X • 0 = 0 • X • 1 = X • Idempotent laws • X + X = X • X • X = X

  3. Basic Theorems (2) • Involution laws • (X’)’ = X • Laws of complementarity • X + X’ = 1 • X • X’ = 0 • Example : • (AB’ + D)E + 1  Anything + 1 = 1 • (AB’ + D)(AB’ + D)’  X • X’ = 0 X

  4. Commutative, Associative Laws • Commutative Laws • XY = YX • X + Y = Y + X • Associative Laws • (XY)Z = X(YZ) • (X + Y) + Z = X + (Y + Z)

  5. Distributive Laws • X(Y + Z) = XY + XZ • X + YZ = (X + Y)(X + Z) • Proof : (X + Y)(X + Z)  X(X + Z) + Y(X + Z)  XX + XZ + YX + YZ  X + XZ + XY + YZ  X (1 + Z + Y) + YZ  X + YZ

  6. Simplification Theorems • XY + XY’ = X • (X+Y)(X + Y’) = X • X + XY = X • X(X + Y) = X • (X + Y’)Y = XY • XY’ + Y = X + Y

  7. Exercise 1 • Simplify the following expression : • Z = A’BC + A’ • Z = [A + B’C + D + EF][A + B’C + (D + EF)’] • Z = (AB + C)(B’D + C’E’) + (AB + C)’

  8. DeMorgan’s Laws • The inverse or complement of any Boolean expression can easily be found by successively applying the following theorems, which are frequently referred to as Demorgan’s Laws • (X + Y)’ = X’Y’ • (XY)’ = X’ + Y’ • Example : find the complement of (A’ + B)C’ [(A’ + B)C’]’  (A’ + B)’ + C  AB’ + C

  9. Example : DeMorgan’s Laws • Find the inverse of F, F = A’B + AB’ F’ = (A’B + AB’)’ = (A’B)’ (AB’)’ = (A’’ + B’)(A’ + B’’) = (A+B’)(A’+B) = AA’ + AB + A’B’ + BB’ = AB + A’B’

  10. Exercise 2 • Find the Boolean expression of F and its truth table and then simplify itand draw the digital circuit equivalent to its simplified expression (1) (2)

  11. TODO • Find the Boolean expression of F and then simplify it and draw the digital circuit equivalent to its simplified expression : • Find the simplified inverse of F and draw its wiring diagram.

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