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.

https://www.gyanodhan.com/video/7B4.%20GATE%20CSEIT/Discrete%20Mathematics%201/398.%20Day%2006Part%2001-%20Discrete%20Mathematics%20for%20Gate%20Hindi%20-%20Group%20theory%20and%20it%27s%20application.mp4

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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

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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।

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2. Group की परिभाषा (Definition of Group)

किसी non-empty set $G$, और एक binary operation $\ast$ पर defined है एक group, यदि वह निम्नलिखित चार conditions (axioms) satisfy करता है:

Group Axioms:

1. Closure: $\forall a,b\in G\Rightarrow a\ast b\in G$

2. Associativity: $\forall a,b,c\in G\Rightarrow (a\ast b)\ast c=a\ast (b\ast c)$

3. Identity Element: $\exists e\in G$ such that $a\ast e=e\ast a=a$

4. Inverse Element: $\forall a\in G$, ∃a−1∈G\exists a^{-1} \in G∃a−1∈G such that a∗a−1=a−1∗a=ea * a^{-1} = a^{-1} * a = ea∗a−1=a−1∗a=e

अगर ये चारों properties satisfy होती हैं, तो set $G$ एक group कहलाता है।

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3. Types of Groups

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4. Example (Addition on Integers)

Let $G=\mathbb{Z}$ (integers), operation: $+$

• Closure: $a+b\in \mathbb{Z}$

• Associativity: $(a+b)+c=a+(b+c)$

• Identity: $0$, since $a+0=a$

• Inverse: $-a$, since $a+(-a)=0$

So, $(\mathbb{Z},+)$ is a group
It is also Abelian

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5. Applications of Group Theory

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Important for GATE:

• Order of a group (number of elements)

• Subgroups

• Lagrange’s Theorem

• Cyclic Groups

• Permutation Groups

• Group Tables

• Application in number theory (modular groups)

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Summary Table:

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