Day 06Part 01- Discrete Mathematics for Gate Hindi – Group theory and it’s application.
Day 06Part 01- Discrete Mathematics for Gate Hindi – Group theory and it’s application.
Would you like a summary of Group Theory and its Applications in Discrete Mathematics for GATE, explained in Hindi? Or are you looking for study materials, video links, or solved problems? Let me know how I can help!
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Day 06 Part 01 – Discrete Mathematics for GATE (in Hindi)
Topic: Group Theory & Its Applications
Language: Hindi-English Mix (for better understanding)
Useful for: GATE, UGC NET, B.Tech (CSE/IT), MCA, BCA, Competitive Exams
1. Group Theory क्या है?
Group Theory एक ऐसा mathematical concept है, जिसमें हम किसी set के elements पर एक operation (जैसे addition, multiplication) apply करते हैं, और check करते हैं कि वे कुछ rules (axioms) satisfy करते हैं या नहीं।
2. Group की परिभाषा (Definition of Group)
किसी non-empty set GG, और एक binary operation ∗* पर defined है एक group, यदि वह निम्नलिखित चार conditions (axioms) satisfy करता है:
Group Axioms:
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Closure: ∀a,b∈G⇒a∗b∈G\forall a, b \in G \Rightarrow a * b \in G
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Associativity: ∀a,b,c∈G⇒(a∗b)∗c=a∗(b∗c)\forall a, b, c \in G \Rightarrow (a * b) * c = a * (b * c)
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Identity Element: ∃e∈G\exists e \in G such that a∗e=e∗a=aa * e = e * a = a
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Inverse Element: ∀a∈G\forall a \in G, ∃a−1∈G\exists a^{-1} \in G such that a∗a−1=a−1∗a=ea * a^{-1} = a^{-1} * a = e
अगर ये चारों properties satisfy होती हैं, तो set GG एक group कहलाता है।
3. Types of Groups
| Type | Condition |
|---|---|
| Abelian Group | यदि group में commutativity हो i.e., a∗b=b∗aa * b = b * a |
| Non-Abelian | यदि a∗b≠b∗aa * b \ne b * a for some a,b∈Ga, b \in G |
| Finite Group | Group with finite number of elements |
| Infinite Group | Group with infinite elements |
4. Example (Addition on Integers)
Let G=ZG = \mathbb{Z} (integers), operation: ++
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Closure: a+b∈Za + b \in \mathbb{Z}
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Associativity: (a+b)+c=a+(b+c)(a + b) + c = a + (b + c)
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Identity: 00, since a+0=aa + 0 = a
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Inverse: −a-a, since a+(−a)=0a + (-a) = 0
So, (Z,+)(\mathbb{Z}, +) is a group It is also Abelian
5. Applications of Group Theory
| Area | Application Example |
|---|---|
| Cryptography | RSA, ECC – modular arithmetic with groups |
| Coding Theory | Error detection & correction |
| Computer Graphics | Symmetry and rotation operations |
| Automata Theory | Transition systems and algebraic structures |
| Compiler Design | Grammar transformations and syntax trees |
Important for GATE:
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Order of a group (number of elements)
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Subgroups
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Lagrange’s Theorem
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Cyclic Groups
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Permutation Groups
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Group Tables
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Application in number theory (modular groups)
Summary Table:
| Property | Meaning |
|---|---|
| Group | Closure, Associative, Identity, Inverse |
| Abelian | Commutative Group |
| Finite Group | Fixed number of elements |
| Applications | Cryptography, Algebra, CS Theory, Graphics |
Would you like:
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GATE-style MCQs with solutions? -
Group table examples? -
A short explainer video in Hindi?
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