Boolean Algebra and Simplification
Why Simplify Boolean Expressions?
Boolean algebra is the mathematics behind every logic circuit. When engineers design chips, they start with a logical requirement and express it as a Boolean expression. Simplifying that expression reduces the number of gates needed, which lowers cost and energy use. Your exam will ask you to simplify expressions step by step, showing the law you used at each stage. Each correctly applied law can earn a mark, so showing your working is as important as the final answer.
The eight laws you need to know are grouped into pairs: each law has a version for AND and a version for OR. Learning them in pairs halves the memorisation effort. The laws use standard Boolean notation: a dot (·) for AND, a plus sign (+) for OR, and an overbar for NOT (written NOT A or A with overbar).
The eight Boolean algebra laws are: Identity, Annulment (Null), Idempotent, Complement, Commutative, Associative, Distributive, and Absorption. Each appears in an AND form and an OR form.
The Eight Laws
The table below lists each law with both its AND and OR forms. Use it as a reference while you work through the examples. Here, A̅ means NOT A (A with overbar).
| Law | AND form | OR form |
|---|---|---|
| Identity | A · 1 = A | A + 0 = A |
| Annulment (Null) | A · 0 = 0 | A + 1 = 1 |
| Idempotent | A · A = A | A + A = A |
| Complement | A · A̅ = 0 | A + A̅ = 1 |
| 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 |
Notice the Absorption law: A + A·B = A. This means that if A is already in the expression, any term that multiplies A by something else is redundant and can be removed. This is the law students most often miss when simplifying — it can collapse several terms into one in a single step.
De Morgan’s Laws
De Morgan’s laws describe how to distribute a NOT across an AND or OR. They are essential for simplifying expressions that contain a NOT over a bracket.
First law: NOT(A AND B) = NOT A OR NOT B. In overbar notation: (A · B) with overbar = A̅ + B̅.
Second law: NOT(A OR B) = NOT A AND NOT B. In overbar notation: (A + B) with overbar = A̅ · B̅.
In plain language: the NOT of an AND equals the OR of the NOTs; the NOT of an OR equals the AND of the NOTs. When you break a NOT over a bracket, the AND becomes OR (or vice versa). De Morgan’s laws are also why NAND and NOR gates are so powerful — a NAND gate directly implements NOT(A AND B).
De Morgan’s laws: (A · B) overbar = A̅ + B̅ and (A + B) overbar = A̅ · B̅. When the NOT is distributed, AND becomes OR and OR becomes AND.
Simplify: A·1 + A·0 + A
Step 1 — Identity law: A·1 = A. Expression becomes: A + A·0 + A.
Step 2 — Annulment law: A·0 = 0. Expression becomes: A + 0 + A.
Step 3 — Identity law (OR form): A + 0 = A. Expression becomes: A + A.
Step 4 — Idempotent law: A + A = A. Final result: A.
In the exam, write the law name beside each step. This tells the marker which rule you are applying and earns partial credit even if you make an arithmetic slip later.
Simplify: A·A̅ + A·B + A
Step 1 — Complement law: A·A̅ = 0. Expression becomes: 0 + A·B + A.
Step 2 — Identity law (OR form): 0 + A·B + A = A·B + A. Expression becomes: A·B + A.
Step 3 — Commutative law: reorder to A + A·B. Step 4 — Absorption law: A + A·B = A. Final result: A.
The Absorption law is the most powerful shortcut. Always scan the expression for a term of the form A + A·(anything) — if you spot one, collapse it to A immediately.
Simplify: NOT(A AND B) OR B — written as (A·B)̅ + B.
Step 1 — De Morgan’s first law: (A·B)̅ = A̅ + B̅. Expression becomes: A̅ + B̅ + B.
Step 2 — Complement law: B̅ + B = 1. Expression becomes: A̅ + 1.
Step 3 — Annulment law (OR form): A̅ + 1 = 1. Final result: 1.
The expression always evaluates to 1, regardless of the inputs. This is a common exam result — when an expression simplifies to a constant, it means a constant-output circuit, which can be replaced by a simple connection to logic HIGH (if output is 1) or logic LOW (if output is 0).