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De Morgan's theorem and duality theorem

Short answer

De Morgan's theorems: (A+B)' = A'·B' and (A·B)' = A'+B'. Duality theorem: interchange AND/OR and 0/1; dual of A+0=A is A·1=A, dual of A+AB=A is A(A+B)=A.

State De Morgan's theorem and Duality theorem with two examples for each.

De Morgan's theorems

  1. The complement of a sum equals the product of the complements: (A + B)' = A'·B'
  2. The complement of a product equals the sum of the complements: (A·B)' = A' + B'

Example 1: (X + Y + Z)' = X'Y'Z'. For X = 1, Y = 0, Z = 0: LHS = (1)' = 0, RHS = 0·1·1 = 0.

Example 2: (A'B)' = A + B'. Check: (A·B)' with A replaced by A' gives (A')' + B' = A + B'.

Duality theorem

Every Boolean identity remains valid if we interchange AND with OR, and 0 with 1 (variables are not changed). The new identity is the dual of the original.

Example 1:

  • A + 0 = A has dual A · 1 = A.
  • A + A' = 1 has dual A · A' = 0.

Example 2:

  • A + AB = A has dual A(A + B) = A.
  • A + A'B = A + B has dual A(A' + B) = AB.

Duality is useful because proving one law automatically proves its dual.

✓ Answer checked by Claude Opus 5.5 (high effort)

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