Boolean Algebra Laws for ISC Class 12 Computer Science
Everything for the Boolean algebra and logic-gate questions of ISC Class 12 Computer Science on one page: the laws and their duals, the gates, universal-gate conversions, adders, minterms and maxterms, K-map rules and the propositional-logic equivalences of the syllabus.
Laws and theorems of Boolean algebra
| Law | Form | Dual form |
|---|---|---|
| Identity | A + 0 = A | A · 1 = A |
| Null (domination) | A + 1 = 1 | A · 0 = 0 |
| Idempotent | A + A = A | A · A = A |
| Complement | A + A' = 1 | A · A' = 0 |
| Involution (double negation) | (A')' = A | (A')' = A |
| Commutative | A + B = B + A | A · B = B · A |
| Associative | A + (B + C) = (A + B) + C | A · (B · C) = (A · B) · C |
| Distributive | A · (B + C) = A · B + A · C | A + B · C = (A + B) · (A + C) |
| Absorption | A + A · B = A | A · (A + B) = A |
| Absorption of the complement | A + A' · B = A + B | A · (A' + B) = A · B |
| De Morgan’s theorem | (A + B)' = A' · B' | (A · B)' = A' + B' |
| Consensus | A · B + A' · C + B · C = A · B + A' · C | (A + B) · (A' + C) · (B + C) = (A + B) · (A' + C) |
The dual of a law swaps AND with OR and 0 with 1. A dash after a letter (A') is NOT.
Logic gates
| Gate | Output | In words |
|---|---|---|
| NOT | Y = A' | inverts the input |
| AND | Y = A · B | 1 only if every input is 1 |
| OR | Y = A + B | 1 if any input is 1 |
| NAND | Y = (A · B)' | AND followed by NOT; a universal gate |
| NOR | Y = (A + B)' | OR followed by NOT; a universal gate |
| XOR | Y = A ⊕ B = A'B + AB' | 1 when the inputs differ |
| XNOR | Y = (A ⊕ B)' = AB + A'B' | 1 when the inputs are equal |
Universal gates
- NOT from NAND: A' = (A · A)'. NOT from NOR: A' = (A + A)'.
- AND from NAND: A · B = ((A · B)')' , that is two NAND gates, the second with its inputs joined.
- OR from NAND: A + B = (A' · B')', that is NAND of the two inverted inputs.
- AND from NOR: A · B = (A' + B')'. OR from NOR: A + B = ((A + B)')'.
Adders
| Circuit | Sum | Carry |
|---|---|---|
| Half adder | S = A ⊕ B | C = A · B |
| Full adder | S = A ⊕ B ⊕ Cin | Cout = A · B + Cin · (A ⊕ B) |
SOP, POS, minterms and maxterms
- A minterm is a product (AND) of every variable, plain where the variable is 1 and complemented where it is 0; a function is the OR of its minterms (SOP, written Σm).
- A maxterm is a sum (OR) of every variable, complemented where the variable is 1 and plain where it is 0; a function is the AND of its maxterms (POS, written ΠM).
- The minterm and maxterm numbers of a function are the rows where it is 1 and where it is 0 respectively.
Karnaugh maps
- A K-map of 2, 3 or 4 variables has 4, 8 or 16 cells; rows and columns follow Gray code (00, 01, 11, 10) so neighbours differ in one variable.
- Group 1s (for SOP) or 0s (for POS) in pairs, quads or octets (1, 2, 4, 8 cells; always a power of two), as large as possible; groups may overlap and may wrap round the edges.
- A variable that changes inside a group is dropped; one that stays the same is kept. Don’t-care cells may be used in a group to make it larger.
Propositional logic
| Law | Equivalence |
|---|---|
| Law of implication | p ⇒ q ≡ ~p ∨ q |
| Law of biconditional | (p ⇔ q) ≡ (p ⇒ q) ∧ (q ⇒ p) |
| Contrapositive | p ⇒ q ≡ ~q ⇒ ~p |
| Converse and inverse | the converse of p ⇒ q is q ⇒ p; the inverse is ~p ⇒ ~q |
| Law of excluded middle | p ∨ ~p ≡ true |
| Law of contradiction | p ∧ ~p ≡ false |
| Law of negation | ~(~p) ≡ p |
| De Morgan’s laws | ~(p ∧ q) ≡ ~p ∨ ~q, and ~(p ∨ q) ≡ ~p ∧ ~q |
| Modus ponens | from p and p ⇒ q, conclude q |
| Chain rule | [(p ⇒ q) ∧ (q ⇒ r)] ⇒ (p ⇒ r) |
Common questions
What is the difference between a minterm and a maxterm?
A minterm is an AND of all the variables, equal to 1 for exactly one row of the truth table; a maxterm is an OR of all the variables, equal to 0 for exactly one row.
Why are NAND and NOR called universal gates?
Any Boolean function can be built using only NAND gates, or only NOR gates, because each of them can make NOT, AND and OR.
What is the duality principle?
Every Boolean law stays true if you swap AND with OR and 0 with 1 everywhere. That is why most laws come in pairs.
How are De Morgan’s theorems stated?
(A + B)' = A' · B' and (A · B)' = A' + B': the complement of a sum is the product of the complements, and the complement of a product is the sum of the complements.
Keep reading
Last checked 2026-10-16.