Boolean Algebra Axioms

Every digital circuit obeys a handful of rules that never break. These rules, called Boolean algebra axioms, are the bedrock of every ALU, decoder, and multiplexer built on silicon.

Core Concept

Boolean algebra works only with two values: 0 (false) and 1 (true). This property is called closure. Every gate output stays within that set, no matter the inputs.

The identity axioms say that ANDing any variable with 1 returns it unchanged, and ORing with 0 leaves it unchanged. The null axioms are the opposite extreme: AND with 0 always gives 0, OR with 1 always gives 1. These four rules explain why unused gate inputs tied to VCC or GND behave predictably.

The complement axiom underlies every inverter. A 74HC04 hex inverter running at 5 V has a propagation delay of about 7 ns and a fan-out of 10 in the 74HC family. Its entire job is producing A' so that A · A' = 0 and A + A' = 1 hold in hardware.

Boolean Expression

The six axioms in compact form: A·1=A, A+0=A, A·0=0, A+1=1, A·A=A, A+A=A, A·A'=0, A+A'=1, (A')'=A. Each axiom has a dual: swap AND with OR and swap 0 with 1 to get the partner axiom. This duality runs through all of Boolean algebra.

Given:
  A = 1, B = 0

Formula / Rule:
  Evaluate F = (A · 1) + (B · A') using axioms

Step by step:
  Step 1: A · 1 = A = 1          (Identity axiom)
  Step 2: A' = 1' = 0           (Complement + Involution)
  Step 3: B · A' = 0 · 0 = 0    (Null axiom: anything AND 0 = 0)
  Step 4: F = 1 + 0 = 1           (Identity axiom: A + 0 = A)

Final Answer:
  F = 1

Exam Tip: GATE often asks which axiom justifies a simplification step. Remember that A+1=1 (not A) — the null axiom for OR always returns 1, not the variable. Also, the dual of every axiom is equally valid; swapping AND/OR and 0/1 always produces another correct axiom. Involution is tested as a trap: applying complement twice returns the original variable.

Key Properties

• Closure: output is always 0 or 1; no intermediate voltage states in ideal logic
• Identity elements: 1 for AND, 0 for OR — distinct values, commonly swapped in exams
• Null elements: 0 for AND, 1 for OR — one input forces the output regardless of others
• Idempotent law holds only in Boolean algebra, not ordinary algebra (2+2 ≠ 2)
• 74HC04 inverter: Vcc = 2–6 V, t_pd ≈ 7 ns at 5 V, fan-out = 10, I_cc ≈ 0.08 mA static
• Involution (double complement) is used in logic minimization to restore a cancelled inversion
• Duality principle: every valid axiom or theorem has a valid dual — swap AND↔OR, 0↔1

Quick Revision

• Six axioms: closure, identity, null, idempotent, complement, involution
• Identity for AND is 1; identity for OR is 0
• Null for AND is 0; null for OR is 1
• A·A'=0 and A+A'=1 always, regardless of what A is
• (A')'=A — double inversion cancels out
• Duality: swap AND↔OR and 0↔1 to get partner axiom
• Exam trap: students write A+1=A instead of A+1=1; the null axiom overrides the variable