How it works
Boolean algebra uses variables that take only two values: 0 and 1. The basic operations are AND (·), OR (+), and NOT (overbar). Key theorems: identity (A+0=A, A·1=A), null (A+1=1, A·0=0), idempotent (A+A=A, A·A=A), complement (A+A'=1, A·A'=0), and involution (A'=A). De Morgan's theorems are essential for NAND/NOR implementation: (AB)' = A'+B' and (A+B)' = A'·B'. Any Boolean expression can be written in Sum of Products (SOP) form by listing minterms, or Product of Sums (POS) by listing maxterms. Simplification by Boolean algebra requires methodical application of theorems — start with complement and absorption laws (A+AB=A, A+A'B=A+B).
Key points to remember
De Morgan's theorem: complement of a product = sum of complements; complement of a sum = product of complements. These are the rules for converting AND-OR logic into NAND-NAND, which uses only one gate type. Duality principle: any Boolean identity remains valid if AND and OR are swapped and 0 and 1 are swapped simultaneously. Canonical SOP uses all variables in each minterm: for f(A,B,C) = Σm(1,3,5,7) — all minterms have A, B, and C present. Absorption law A+AB = A and its dual A(A+B) = A are frequently tested simplification steps. XOR identity: A⊕B = A'B + AB'; XNOR is its complement. The consensus theorem AB + A'C + BC = AB + A'C often appears in simplification problems.
Exam tip
The examiner always asks you to apply De Morgan's theorem to convert an expression into NAND-NAND form or to prove a Boolean identity — write both De Morgan forms clearly and apply them step by step without skipping intermediate expressions.