A complete interactive guide — from first principles to De Morgan's theorems, with worked examples and graded exercises.
Boolean Algebra is a branch of algebra where every variable takes exactly one of two values: 0 (false) and 1 (true). Developed by George Boole in 1854, it provides the mathematical foundation for all digital logic — every gate, register, and processor is built from Boolean operations.
In OCR H446 Component 1, you must be able to read and write Boolean expressions, construct truth tables, simplify logic circuits, apply Boolean laws, and use De Morgan's theorems to transform expressions.
· means AND, addition + means OR, and a bar over a variable (e.g. Ā) means NOT. These symbols have nothing to do with arithmetic.
Venn diagrams are used throughout this guide — each circle represents an input variable, the rectangle represents the universal set U (all possible inputs), and shaded regions show where the output is 1.
| Symbol | Name | Meaning | Example |
|---|---|---|---|
| · | AND | Both inputs must be 1 (true) | A · B = 1 only when A=1 AND B=1 |
| + | OR | At least one input must be 1 (true) | A + B = 1 when A=1 OR B=1 (or both) |
| ¬ or A' | NOT | Reverses the value (0 becomes 1, 1 becomes 0) | ¬A = 1 when A=0 |
| ⊕ | XOR | Exactly one input must be 1 (not both) | A ⊕ B = 1 when A≠B |
Every Boolean expression is built from these fundamental operations. Memorise their symbols, truth conditions, and Venn representations — they appear throughout all simplification and circuit questions.
Select an operation below, then toggle inputs A and B to see which regions are shaded and how the output changes. The highlighted row in the truth table shows the current input combination.
These laws are the toolkit for every simplification. OCR examiners expect you to name the law at each step — not just write the result. Study the AND and OR forms together; they are duals of each other.
| Law | AND form | OR form |
|---|---|---|
| Identity | A · 1 = A | A + 0 = A |
| Null / Dominance | A · 0 = 0 | A + 1 = 1 |
| Idempotent | A · A = A | A + A = A |
| Complement | A · Ā = 0 | A + Ā = 1 |
| Double Negation | A̿ = 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 |
| De Morgan 1 | ̄(A · B) = Ā + B̄ | |
| De Morgan 2 | ̄(A + B) = Ā · B̄ | |
De Morgan's theorems are the most frequently examined laws in Boolean simplification. They allow you to convert between NAND/NOR forms and AND/OR/NOT forms — essential when working with logic circuits.
De Morgan extends to any number of variables. For three variables: ̄(A·B·C) = Ā + B̄ + C̄ and ̄(A+B+C) = Ā · B̄ · C̄. The rule is the same — break the bar, flip every operator.
By De Morgan, ̄(A·A) = Ā + Ā = Ā — so a NAND gate with both inputs tied together acts as NOT. And ̄(̄A · ̄B) = A + B — so two NOTs into a NAND gives OR. Any Boolean function can therefore be expressed using only NAND gates. NOR is functionally complete for the same reason.
Click any card to expand the full step-by-step solution. Each step names the law used — replicate this in your exam answers.
Attempt each question before revealing the solution. A hint is provided for each — use it only if you're stuck after a genuine attempt.
Marks in simplification questions are awarded for each correct step with its justification. Writing the final answer alone scores zero. Name the law (e.g. "De Morgan's theorem", "absorption", "complement law") at every step.
With n input variables there are exactly 2ⁿ rows. List combinations in binary counting order (00, 01, 10, 11 for two variables) to guarantee completeness. One missing row costs the mark for that row and can invalidate your final column.
When given a circuit diagram, write its Boolean expression before attempting simplification. Work output-to-input, gate by gate. Only then apply algebraic laws.
Functional completeness of NAND and NOR is a favourite short-answer topic. State clearly: (1) NAND can implement NOT (tie inputs), AND (NAND then NOT), and OR (De Morgan). (2) Since AND, OR, NOT are sufficient for all Boolean functions, NAND alone is sufficient.
When asked to prove two expressions are equal, manipulate one side only until it matches the other. Never manipulate both sides simultaneously — that is not a valid proof method.
XOR is tested in the context of arithmetic circuits. A half adder produces a Sum = A ⊕ B and a Carry = A · B. Know this combination — it connects Boolean Algebra directly to binary addition.