Unit 11.1A · Term 1

Boolean Expressions & Simplification

Boolean algebra uses laws and rules to simplify complex logical expressions. Simpler expressions mean fewer gates, smaller circuits, and faster computers.

Learning Objectives

11.3.3.1 Understand laws of Boolean logic
11.3.3.2 Simplify logical expressions

Lesson Presentation

11.1A-boolean-expressions.pdf · Slides for classroom use

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Conceptual Anchor

The Algebra Parallel

Boolean algebra works like regular algebra but with 0 and 1 instead of numbers. Just as x + 0 = x in regular maths, A OR 0 = A in Boolean. The same simplification principles apply — factor, combine, reduce.

Rules & Theory

Boolean Algebra Laws

Law	AND form	OR form
Identity	A · 1 = A	A + 0 = A
Null	A · 0 = 0	A + 1 = 1
Idempotent	A · A = A	A + A = A
Complement	A · Ā = 0	A + Ā = 1
Double Negation	NOT(NOT 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

De Morgan's Theorem

De Morgan's Laws (crucial for simplification):

  NOT(A AND B) = (NOT A) OR (NOT B)
  NOT(A OR B)  = (NOT A) AND (NOT B)

In symbols:
  (A · B)' = A' + B'     ← "break the bar, change the sign"
  (A + B)' = A' · B'

Memory Trick for De Morgan's

"Break the bar, change the sign" — when you move a NOT bar over a group, the NOT distributes to each variable and AND becomes OR (and vice versa).

Worked Examples

1 Simplify: A · B + A · C

  A · B + A · C
= A · (B + C)         ← Distributive law (factor out A)

Result: A · (B + C)   ← uses 2 gates instead of 3

2 Simplify: A + A · B

  A + A · B
= A                    ← Absorption law

Proof with truth table:
 A  B │ A·B │ A + A·B │ A
─────┼─────┼────────┼───
 0  0 │  0  │   0    │ 0
 0  1 │  0  │   0    │ 0
 1  0 │  0  │   1    │ 1
 1  1 │  1  │   1    │ 1
                           All match! ✓

3 Apply De Morgan's: NOT(A OR B)

  NOT(A OR B)
= NOT(A) AND NOT(B)    ← De Morgan's Law
= A' · B'

Common Pitfalls

NOT Distributes Over AND/OR

NOT(A AND B) ≠ NOT(A) AND NOT(B). You must use De Morgan's Law — NOT distributes AND the operator changes: NOT(A AND B) = NOT(A) OR NOT(B).

Tasks

Remember

State the Identity, Null, and Complement laws for AND and OR.

Apply

Simplify: (A · B) + (A · B') using Boolean laws.

Apply

Apply De Morgan's theorem to simplify: NOT(X AND Y AND Z).

Analyze

Prove using a truth table that A · (A + B) = A (absorption law).

Self-Check Quiz

Q1: What does the Absorption law state?

A + A·B = A and A·(A+B) = A. The extra term is "absorbed".

Q2: Apply De Morgan's to NOT(A AND B).

NOT(A) OR NOT(B) — break the bar, change AND to OR.

Q3: Simplify A · 1 + A · 0.

A · 1 + A · 0 = A + 0 = A (Identity + Null laws).