Siksha Sarovar

Siksha Sarovar (sikshasarovar.com) is a free educational web application that helps students in India learn programming and prepare for academic and competitive exams. The platform offers structured coding courses (C, C++, Python, Java, HTML, CSS, PHP, Power BI, AI, Machine Learning, Data Science), complete university curriculum notes for BCA/MCA students with previous year question papers, Class 10 and Class 12 CBSE/HBSE school notes, and dedicated preparation material for SSC, UPSC, Banking, Railway and other government exams. Browsing the site is completely free and requires no account. Users may optionally sign in with Google solely to save their learning progress, quiz scores and personal preferences across devices.

Privacy Policy | Terms of Service | Contact Siksha Sarovar | About Siksha Sarovar

v4.0.9 · PWA
Siksha Sarovar logo
Siksha Sarovar
Your Learning Universe

Siksha Sarovar is a free e-learning platform for coding courses, BCA university notes and competitive exam preparation. Optional Google sign-in saves your learning progress across devices.

Initializing knowledge base…
Compiling modules 0%

Unit I Overview: Boolean Algebra, Logic Gates and Arithmetic Circuits

Lesson 2 of 17 in the free Computer Organization and Architecture notes on Siksha Sarovar, written by Rohit Jangra.

Unit I: Boolean Algebra, Logic Gates and Arithmetic Circuits

Unit I is the digital foundation of every CPU, memory chip, and I/O controller. Mastering it unlocks understanding of all subsequent hardware topics.

---

---

Unit I Lessons Overview

LessonTitleKey Topics
Lesson 3Logic Gates and Boolean SimplificationAND, OR, NOT, NAND, NOR, XOR, XNOR; SOP/POS; Minterms
Lesson 4Karnaugh Maps2/3/4-variable K-maps; Grouping rules; Don't care; Prime implicants
Lesson 5Arithmetic CircuitsHalf/Full Adder; Subtractors; Ripple Carry Adder; CLA

---

Complete Boolean Algebra Laws

LawAND FormOR Form
IdentityA · 1 = AA + 0 = A
Null (Annihilator)A · 0 = 0A + 1 = 1
IdempotentA · A = AA + A = A
ComplementA · A' = 0A + A' = 1
Double Negation(A')'= A(A')'= A
CommutativeA · B = B · AA + B = B + A
Associative(A·B)·C = A·(B·C)(A+B)+C = A+(B+C)
DistributiveA·(B+C) = A·B + A·CA+(B·C) = (A+B)·(A+C)
AbsorptionA·(A+B) = AA + A·B = A
De Morgan(A·B)' = A' + B'(A+B)' = A' · B'

---

De Morgan's Theorem — Truth Table Proof

ABA·B(A·B)'A'B'A'+B'
0001111
0101101
1001011
1110000

Columns (A·B)' and A'+B' are identical → De Morgan's First Theorem proved. ✓

ABA+B(A+B)'A'B'A'·B'
0001111
0110100
1010010
1110000

Columns (A+B)' and A'·B' are identical → De Morgan's Second Theorem proved. ✓

---

Study Deep: Boolean Minimization Reduces IC Cost

Minimizing Boolean expressions reduces gate count, which:

  • Reduces die area — fewer transistors per chip
  • Lowers power consumption — fewer switching events
  • Improves speed — fewer gate delays on the critical path
  • Cuts cost — smaller chip, higher yield per wafer

Example: F = A'B'C + A'BC + AB'C + ABC simplifies to F = C (all other variables cancel). This reduces 12 gates to 0 gates!

📝 Exam Tip: De Morgan's theorem is tested every exam. Rule: break the bar, change the operator. (A·B)' = A' + B'. NAND and NOR are universal gates — any logic circuit can be built using only NAND gates or only NOR gates.