Boolean Algebra and Logic Gates
This domain tests the algebra of 0 and 1, truth tables, gate behaviour, canonical forms and simplification by identities and Karnaugh maps.
Core identities
- A + 0 = A, A · 1 = A, A + 1 = 1, A · 0 = 0
- A + A = A, A · A = A, A + A' = 1, A · A' = 0
- Absorption: A + AB = A, A(A + B) = A
- De Morgan: (A + B)' = A' · B' and (A · B)' = A' + B'
NAND and NOR are universal gates: either one alone can build AND, OR and NOT. XOR gives 1 only when inputs differ and is not universal.
Canonical forms
A minterm is a product term containing every variable once. Sum of minterms (SOP) is read from the rows where output = 1; product of maxterms (POS) from rows where output = 0.
Worked example
Simplify F = AB + A'B + AB'.
- AB + A'B = B(A + A') = B
- So F = B + AB' = (B + A)(B + B') = A + B Same result by absorption: X + X'Y = X + Y, but here direct factoring is safer. Final answer: F = A + B.
K-map notes
Group 1s in blocks of 1, 2, 4, 8…; groups may wrap around edges; make groups as large as possible; a group's term keeps only the variables that do not change inside it.
Traps
Applying De Morgan without switching the operator; grouping three 1s in a K-map; forgetting don't-cares can be used as 1 or 0 to enlarge a group; calling XOR a universal gate.