Boolean Algebra and Logic Gates
Identities and simplification
Candidates must know the standard Boolean identities: commutative, associative, distributive, identity (A+0=A, A·1=A), null (A+1=1, A·0=0), idempotent (A+A=A), complement (A+A'=1, A·A'=0), absorption (A+AB=A), and De Morgan's laws: (A+B)' = A'·B' and (A·B)' = A'+B'. Be able to simplify expressions and convert between sum-of-products (SOP) and product-of-sums (POS) forms, and to read a truth table into a canonical expression.
Gates and universality
Know the truth tables for AND, OR, NOT, NAND, NOR, XOR, and XNOR. NAND and NOR are universal - any function can be built from either alone. XOR detects difference and is the basis of parity and adder sum logic; XNOR detects equality.
Traps to watch
- Applying De Morgan incorrectly by forgetting to flip the operator as well as the inputs.
- Confusing XOR with OR: XOR is 1 only when inputs differ.
- Assuming a circuit is minimal when it is merely correct - Karnaugh maps and algebraic simplification reduce gate count.
- Mixing up SOP (AND of OR terms? no - OR of AND terms) and POS.
- Forgetting that a NAND is NOT-AND, so its output is 0 only when both inputs are 1.
Rule of thumb
NAND output = 0 only for all-ones inputs; NOR output = 1 only for all-zero inputs. Use these two facts to evaluate circuits quickly.
Worked example: Simplify F = A·B + A·B'. Factor A: A·(B + B') = A·1 = A. So the two-term circuit collapses to a single wire from A - a classic absorption/complement result.