Digital Electronics/ 2-valued Boolean algebra and it’s theorem in Digital Logic with easy explanation.
Digital Electronics/ 2-valued Boolean algebra and it’s theorem in Digital Logic with easy explanation.
Contents
- 0.1 2-Valued Boolean Algebra & Its Theorems in Digital Logic
- 0.2 🔹 What is 2-Valued Boolean Algebra?
- 0.3 Basic Boolean Operations
- 0.4 1️⃣ AND Operation (⋅)
- 0.5 2️⃣ OR Operation (+)
- 0.6 3️⃣ NOT Operation (¬ or ‘)
- 0.7 Boolean Algebra Theorems
- 0.8 1️⃣ Identity Law
- 0.9 2️⃣ Null Law
- 0.10 3️⃣ Idempotent Law
- 0.11 4️⃣ Involution Law
- 0.12 5️⃣ Complement Law
- 0.13 6️⃣ Distributive Law
- 0.14 7️⃣ Absorption Law
- 0.15 8️⃣ De Morgan’s Theorems
- 0.16 Example: Simplify the Expression
- 0.17 Summary
- 0.18 Digital Electronics/ 2-valued Boolean algebra and it’s theorem in Digital Logic with easy explanation.
- 0.19 01. Boolean Algebra and Logic Gates.pmd
- 0.20 UNIT-II Boolean algebra and Logic Gates
- 0.21 Boolean Algebra and Logic Gate
- 1 💡 What is 2-Valued Boolean Algebra?
- 2 ⚙️ Basic Boolean Operations
- 3 📘 Basic Boolean Theorems / Laws
- 4 🧠 Tip to Remember
- 5 🔧 Real-Life Application
- 6 📝 Summary Cheat-Sheet
- 7 📥 Want a PDF or Video Explanation?
2-Valued Boolean Algebra & Its Theorems in Digital Logic
🔹 What is 2-Valued Boolean Algebra?
Boolean Algebra is a mathematical system used in digital electronics where variables take only two values:
✅ 0 (False / LOW)
✅ 1 (True / HIGH)
All digital circuits (AND, OR, NOT gates) use Boolean algebra to simplify logic expressions.
Basic Boolean Operations
1️⃣ AND Operation (⋅)
- Symbol: A⋅BA \cdot B or ABAB
- Truth Table:
A B A ⋅ B 0 0 0 0 1 0 1 0 0 1 1 1 - Example: 1⋅1=1,1 \cdot 1 = 1, 0⋅1=00 \cdot 1 = 0
2️⃣ OR Operation (+)
- Symbol: A+BA + B
- Truth Table:
A B A + B 0 0 0 0 1 1 1 0 1 1 1 1 - Example: 1+0=1,1 + 0 = 1, 0+0=00 + 0 = 0
3️⃣ NOT Operation (¬ or ‘)
- Symbol: A′A’ or ¬A\neg A
- Truth Table:
A A’ 0 1 1 0 - Example: 1′=0,1′ = 0, 0′=10′ = 1
Boolean Algebra Theorems
These theorems help in simplifying logic circuits.
1️⃣ Identity Law
- A+0=AA + 0 = A
- A⋅1=AA \cdot 1 = A
2️⃣ Null Law
- A+1=1A + 1 = 1
- A⋅0=0A \cdot 0 = 0
3️⃣ Idempotent Law
- A+A=AA + A = A
- A⋅A=AA \cdot A = A
4️⃣ Involution Law
- (A′)′=A(A’)’ = A
5️⃣ Complement Law
- A+A′=1A + A’ = 1
- A⋅A′=0A \cdot A’ = 0
6️⃣ Distributive Law
- A⋅(B+C)=(A⋅B)+(A⋅C)A \cdot (B + C) = (A \cdot B) + (A \cdot C)
- A+(B⋅C)=(A+B)⋅(A+C)A + (B \cdot C) = (A + B) \cdot (A + C)
7️⃣ Absorption Law
- A+(A⋅B)=AA + (A \cdot B) = A
- A⋅(A+B)=AA \cdot (A + B) = A
8️⃣ De Morgan’s Theorems
- (A⋅B)′=A′+B′(A \cdot B)’ = A’ + B’
- (A+B)′=A′⋅B′(A + B)’ = A’ \cdot B’
Example: Simplify the Expression
Expression: A+(A⋅B)A + (A \cdot B)
Using Absorption Law:
A+(A⋅B)=AA + (A \cdot B) = A
So, instead of using a circuit with AND and OR gates, we can directly use A, saving hardware.
Summary
✔ Boolean Algebra is used in Digital Logic & Circuits.
✔ 2-Valued Boolean Algebra uses only 0 & 1.
✔ Boolean Theorems help simplify logic circuits.
Would you like circuit diagrams or more solved examples?
Digital Electronics/ 2-valued Boolean algebra and it’s theorem in Digital Logic with easy explanation.
01. Boolean Algebra and Logic Gates.pmd
UNIT-II Boolean algebra and Logic Gates
Boolean Algebra and Logic Gate
Here’s a clear and easy-to-understand explanation of 2-Valued Boolean Algebra and its theorems in Digital Electronics / Digital Logic, ideal for students of B.Tech, GATE, or competitive exams.
💡 What is 2-Valued Boolean Algebra?
Boolean Algebra is a mathematical structure used to perform operations on binary variables (0 and 1).
In 2-valued Boolean algebra, each variable has only two possible values:
-
0
→ False/Low -
1
→ True/High
It forms the backbone of digital circuits and logic design.
⚙️ Basic Boolean Operations
Operation | Symbol | Meaning | Example |
---|---|---|---|
AND | · or ∧ | Multiply | 1·1 = 1 |
OR | + | Add | 1+0 = 1 |
NOT | ¬ or ‘ | Invert | ¬1 = 0 |
📘 Basic Boolean Theorems / Laws
✅ 1. Identity Laws
-
A + 0 = A
-
A · 1 = A
✅ 2. Null Laws
-
A + 1 = 1
-
A · 0 = 0
✅ 3. Idempotent Laws
-
A + A = A
-
A · A = A
✅ 4. Complement Laws
-
A + A’ = 1
-
A · A’ = 0
✅ 5. Double Negation Law
-
(A’)’ = A
✅ 6. Commutative Laws
-
A + B = B + A
-
A · B = B · A
✅ 7. Associative Laws
-
A + (B + C) = (A + B) + C
-
A · (B · C) = (A · B) · C
✅ 8. Distributive Laws
-
A · (B + C) = A·B + A·C
-
A + (B · C) = (A + B) · (A + C)
✅ 9. Absorption Laws
-
A + A·B = A
-
A · (A + B) = A
✅ 10. De Morgan’s Theorems
-
(A · B)’ = A’ + B’
-
(A + B)’ = A’ · B’
💡 These are extremely important in circuit simplification.
🧠 Tip to Remember
Use truth tables to verify the theorems. For example:
Theorem: A + A’ = 1
A | A’ | A + A’ |
---|---|---|
0 | 1 | 1 |
1 | 0 | 1 |
✅ Verified!
🔧 Real-Life Application
All logic gates (AND, OR, NOT, NAND, NOR, XOR) are physical implementations of Boolean operations.
Using these laws:
-
We simplify digital circuits
-
We design efficient hardware
-
We minimize logic expressions
📝 Summary Cheat-Sheet
Law/Rule Name | Expression(s) |
---|---|
Identity | A + 0 = A, A · 1 = A |
Null | A + 1 = 1, A · 0 = 0 |
Idempotent | A + A = A, A · A = A |
Complement | A + A’ = 1, A · A’ = 0 |
Double Negation | (A’)’ = A |
Commutative | A + B = B + A, A · B = B · A |
Associative | A + (B + C) = (A + B) + C |
Distributive | A · (B + C) = A·B + A·C |
Absorption | A + A·B = A |
De Morgan’s Laws | (A · B)’ = A’ + B’, (A + B)’ = A’ · B’ |
📥 Want a PDF or Video Explanation?
I can generate:
-
A 1-page PDF summary (perfect for revision)
-
A video script for YouTube/Tutorial
-
A truth table practice worksheet
Just let me know what format you prefer!