Boolean Algebra and Karnaugh Maps

• Boolean Relations
• Duality Relations
• Sum of Products
• Algebraic Simplifications
• Karnaugh Maps:
  • Pairs, Quads, Octets
  • Simplifications:
    • Overlapping
    • Redundant
    • Rolling
• Don't care outputs

Boolean Relations

• Commutative, Associative, and Distributive
• OR:
  • A + 0 = ?
  • A + A = ?
  • A + 1 = ?
• AND:
  • A . 0 = ?
  • A . A = ?
  • A . 1 = ?
• double negation
• De Morgan's theorems

Duality Relations

1. change ORs to ANDs
2. change ANDs to ORs
3. complement constants (0's and 1's)

• example: apply to A(B + C) = AB + AC
• similar to negative logic

Sum of Products

• given a truth table, we want to design a circuit
• four fundamental products of two inputs:
  1. Y = (bar(A)) (bar(B))
  2. Y = (bar(A)) B
  3. Y = A (bar(B))
  4. Y = A B
• truth table ("1-output" rows) gives us the fundamental products that produce a high output -- OR these
• example: XOR = A (bar(B)) | (bar(A)) B
• works for any truth table

Algebraic Simplification

• minimize the number of gate inputs
• example: Y = A (bar(B)) + A B
• example: Y = A B + A C + B D + C D
• in-class exercise: simplify Y = (bar(A)) (bar(B)) C D + bar {A} B C D + A (bar(B)) bar {C} D

Karnaugh Maps

• Complement and direct on rows, column

• Write 1 for the high outputs, 0 for the rest

Pairs, Quads, Octets

• try and group pairs of ones -- the corresponding (direct and complement) variable can be eliminated
• try and group quads of ones -- the corresponding (direct and complement) variables can be eliminated
• try and group octets of ones

Simplifications

• overlapping (two pairs, quads, octets using the same "1") is fine
• can always non-overlap, but may lead to more complicated equations
• can go "over the edge" of the map ("rolling" the map)
• eliminate redundant groups

Don't care Outputs

• X marks a "don't care" output
• truth table
• what groupings would you do for this map? (bigger groups are always better).
• how would you implement it using gates?