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.

 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

AND Operation (⋅)

• Symbol: $A\cdot B$ or $AB$
• Truth Table:
  

  A	B	A ⋅ B
  0	0	0
  0	1	0
  1	0	0
  1	1	1

• Example: $1\cdot 1=1,$ $0\cdot 1=0$

OR Operation (+)

• Symbol: $A+B$
• Truth Table:
  

  A	B	A + B
  0	0	0
  0	1	1
  1	0	1
  1	1	1

• Example: $1+0=1,$ $0+0=0$

NOT Operation (¬ or ‘)

• Symbol: A′A’A′ or $\neg A$
• Truth Table:
  

  A	A’
  0	1
  1	0

• Example: 1′=0,1′ = 0,1′=0, 0′=10′ = 10′=1

 Boolean Algebra Theorems

These theorems help in simplifying logic circuits.

Identity Law

• $A+0=A$
• $A\cdot 1=A$

Null Law

• $A+1=1$
• $A\cdot 0=0$

Idempotent Law

• $A+A=A$
• $A\cdot A=A$

Involution Law

• (A′)′=A(A’)’ = A(A′)′=A

Complement Law

• A+A′=1A + A’ = 1A+A′=1
• A⋅A′=0A \cdot A’ = 0A⋅A′=0

Distributive Law

• $A\cdot (B+C)=(A\cdot B)+(A\cdot C)$
• $A+(B\cdot C)=(A+B)\cdot (A+C)$

Absorption Law

• $A+(A\cdot B)=A$
• $A\cdot (A+B)=A$

De Morgan’s Theorems

• (A⋅B)′=A′+B′(A \cdot B)’ = A’ + B’(A⋅B)′=A′+B′
• (A+B)′=A′⋅B′(A + B)’ = A’ \cdot B’(A+B)′=A′⋅B′

 Example: Simplify the Expression

Expression: $A+(A\cdot B)$
Using Absorption Law:
$A+(A\cdot B)=A$

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.

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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.

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Basic Boolean Operations

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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

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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)

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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.

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Tip to Remember

Use truth tables to verify the theorems. For example:

Theorem: A + A’ = 1

Verified!

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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

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Summary Cheat-Sheet

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Digital Electronics/ 2-valued Boolean algebra and it’s theorem in Digital Logic with easy explanation.

DIGITAL ELECTRONICS BOOLEAN ALGEBRA

Digital Electronics (18EC32) Notes