Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least.
Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least.
Contents [hide]
- 0.1 Introduction to Boolean Algebra
- 0.2 Boolean Algebra के महत्वपूर्ण गुण (Properties of Boolean Algebra)
- 0.3 Closure Property (सम्पूर्णता गुण)
- 0.4 Identity Property (परिचायक गुण)
- 0.5 Commutative Property (अविनियम गुण)
- 0.6 Distributive Property (वितरण गुण)
- 0.7 Complement Property (परिपूरक गुण)
- 0.8 Summary (सारांश)
- 0.9 Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least.
- 0.10 BOOLEAN ALGEBRA 2.1 Introduction
- 0.11 3 Boolean Algebra – I Love Maths
- 1
Boolean Algebra – Introduction - 2
Key Properties in Boolean Algebra- 2.1
Closure Property - 2.2
Identity Property - 2.3
Commutative Property - 2.4
Distributive Property - 2.5
Complement Law - 2.6
Bonus: Other Useful Laws - 2.7 Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least.
- 2.8 UNIT – 1 NUMBER SYSTEMS & BOOLEAN ALGEBRA
- 2.9 LATTICE AND BOOLEAN ALGEBRA
- 2.1
Introduction to Boolean Algebra
Boolean Algebra एक Mathematical System है जिसे George Boole ने विकसित किया था। यह Binary System (0 और 1) पर आधारित होता है और इसका उपयोग Digital Logic, Circuit Design, and Computer Science में किया जाता है।
Basic Operations in Boolean Algebra
Boolean Algebra में तीन मुख्य ऑपरेशंस होते हैं:
- AND (⋅) → Output 1 तभी होता है जब दोनों Inputs 1 हों।
- OR (+) → Output 1 होता है अगर कोई भी Input 1 हो।
- NOT ( ̅ ) → Input का उल्टा Output देता है (0 → 1 और 1 → 0)।
Boolean Algebra के महत्वपूर्ण गुण (Properties of Boolean Algebra)
Closure Property (सम्पूर्णता गुण)
Boolean Algebra में किसी भी दो Boolean वैरिएबल्स को AND (+) या OR (⋅) करने पर भी Boolean वैल्यू (0 या 1) ही मिलती है।
यदि A और B Boolean Variables हैं, तो:
- A + B (OR Operation) = Boolean Value (0 या 1)
- A ⋅ B (AND Operation) = Boolean Value (0 या 1)
Example:
- 1 + 0 = 1 (OR operation)
- 1 ⋅ 0 = 0 (AND operation)
Boolean Algebra Closed रहता है क्योंकि यह हमेशा 0 और 1 के दायरे में रहता है।
Identity Property (परिचायक गुण)
Identity Property बताती है कि 0 और 1, OR और AND ऑपरेशन में कैसे व्यवहार करते हैं।
For AND (⋅) Operation:
- A ⋅ 1 = A
- A ⋅ 0 = 0
For OR (+) Operation:
- A + 0 = A
- A + 1 = 1
Example:
- 1 ⋅ 1 = 1
- 1 + 0 = 1
- 0 ⋅ 1 = 0
1 और 0 की यह विशेषता Boolean Algebra में पहचान (Identity) बनाए रखती है।
Commutative Property (अविनियम गुण)
Commutative Property कहती है कि Boolean Algebra में OR और AND के लिए Operands की Order बदलने से Result पर कोई फर्क नहीं पड़ता।
For AND (⋅) Operation:
- A ⋅ B = B ⋅ A
For OR (+) Operation:
- A + B = B + A
Example:
- 1 + 0 = 0 + 1 = 1
- 1 ⋅ 0 = 0 ⋅ 1 = 0
Commutative Property Boolean Operations को स्वतंत्रता देती है।
Distributive Property (वितरण गुण)
Distributive Property यह बताती है कि किसी Boolean Expression में OR और AND कैसे वितरित (distribute) होते हैं।
AND over OR:
- A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C)
OR over AND:
- A + (B ⋅ C) = (A + B) ⋅ (A + C)
Example:
- 1 ⋅ (0 + 1) = (1 ⋅ 0) + (1 ⋅ 1)
- = 0 + 1 = 1 (LHS = RHS)
Distributive Property Boolean Expressions को Factorization और Expansion करने में मदद करती है।
Complement Property (परिपूरक गुण)
Complement Property यह बताती है कि किसी Boolean Variable का उल्टा करने (NOT Operation) से क्या होता है।
Rules:
- A + A̅ = 1
- A ⋅ A̅ = 0
- (A̅)̅ = A (Double Complement gives the original value)
Example:
- अगर A = 1, तो A̅ = 0
- A + A̅ = 1 + 0 = 1
- A ⋅ A̅ = 1 ⋅ 0 = 0
Complement Property Boolean Functions को Simplify करने में मदद करती है।
Summary (सारांश)
| Property | Expression | Example |
|---|---|---|
| Closure | A + B, A ⋅ B ∈ {0,1} | 1 + 0 = 1 |
| Identity | A + 0 = A, A ⋅ 1 = A | 1 ⋅ 1 = 1 |
| Commutative | A + B = B + A, A ⋅ B = B ⋅ A | 0 + 1 = 1 + 0 |
| Distributive | A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C) | 1 ⋅ (0 + 1) = (1 ⋅ 0) + (1 ⋅ 1) |
| Complement | A + A̅ = 1, A ⋅ A̅ = 0 | 1 + 0 = 1 |
Boolean Algebra का उपयोग Digital Circuits, Logic Gates, and Computer Algorithms को Design करने में किया जाता है।
क्या आप Boolean Algebra से जुड़े किसी और Concept को विस्तार से समझना चाहते हैं?
Boolean algebra introduction- closed Identity, Commutative, Distributive, Complement At least.
BOOLEAN ALGEBRA 2.1 Introduction
3 Boolean Algebra – I Love Maths
Here’s a concise introduction to Boolean Algebra with key properties like Closure, Identity, Commutative, Distributive, and Complement — essential for understanding digital logic and computer science fundamentals.
Boolean Algebra – Introduction
Boolean algebra deals with binary values:
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0 (False)
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1 (True)
Using logical operations: -
AND (·)
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OR (+)
-
NOT (‘)
Key Properties in Boolean Algebra
Closure Property
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Boolean algebra is closed under AND, OR, and NOT.
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Example: If A, B ∈ {0,1}, then A + B ∈ {0,1}, A · B ∈ {0,1}
Identity Property
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For OR: A + 0 = A
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For AND: A · 1 = A
Commutative Property
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A + B = B + A
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A · B = B · A
Order of operation doesn’t matter.
Distributive Property
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AND distributes over OR: A · (B + C) = A·B + A·C
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OR distributes over AND: A + (B · C) = (A + B) · (A + C)
Complement Law
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A + A’ = 1
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A · A’ = 0
Where A’ is the complement (NOT A)
Bonus: Other Useful Laws
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Idempotent: A + A = A, A · A = A
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Domination: A + 1 = 1, A · 0 = 0
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Double Negation: (A’)’ = A
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