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Day 02-Discrete mathematics for computer science in Hindi – Type of Relation with basic concept

Ali Seron Jason Ali Seron Jason

Day 02-Discrete mathematics for computer science in Hindi – Type of Relation with basic concept

https://www.gyanodhan.com/video/7B5.%20GATE%20CSEIT/Discrete%20Mathematics%202/531.%20Day%2002-Discrete%20mathematics%20for%20computer%20science%20in%20Hindi%20-%20Type%20of%20Relation%20with%20basic%20concept.mp4

बिलकुल! यह है Day 02 का पूरा नोट्स और समझाया हुआ भाग —
Discrete Mathematics for Computer Science (CSE/IT) in Hindi

🔹 टॉपिक: Types of Relations (संबंध के प्रकार) और उसका बेसिक कॉन्सेप्ट

📘 रिलेशन (Relation) क्या होता है?

अगर हमारे पास दो sets हों, A और B, तो उनका Cartesian Product:

A×B={(a,b) ∣ a∈A,b∈B}A \times B = \{ (a, b) \ | \ a \in A, b \in B \}

अब, इस Cartesian Product का कोई subset कहलाता है एक Relation (R)

Relation: A से B तक किसी नियम या कंडीशन पर आधारित ordered pairs का collection

Types of Relations (संबंधों के प्रकार)

क्र.सं. प्रकार (Type) परिभाषा (Definition in Hindi)
1⃣ Reflexive Relation हर element खुद से related हो: (a,a)∈R(a, a) \in R
2⃣ Symmetric Relation अगर (a,b)∈R(a, b) \in R है तो (b,a)∈R(b, a) \in R भी होना चाहिए
3⃣ Anti-Symmetric Relation अगर (a,b)∈R(a, b) \in R और (b,a)∈R(b, a) \in R, तो a=ba = b होना चाहिए
4⃣ Transitive Relation अगर (a,b)∈R(a, b) \in R और (b,c)∈R(b, c) \in R, तो (a,c)∈R(a, c) \in R भी हो
5⃣ Equivalence Relation जो Reflexive + Symmetric + Transitive हो
6⃣ Irreflexive Relation कोई भी element खुद से related न हो: (a,a)∉R(a, a) \notin R

1. Reflexive Relation (प्रत्यावर्ती संबंध)

Definition: (a,a)∈R(a, a) \in R ∀ a ∈ A
Example: A = {1,2}, R = {(1,1), (2,2)} — Reflexive

2. Symmetric Relation (साम्य संबंध)

Definition: (a,b)∈R⇒(b,a)∈R(a, b) \in R \Rightarrow (b, a) \in R
Example: {(1,2), (2,1)}
Not symmetric: {(1,2)} only

3. Anti-Symmetric Relation (प्रति-साम्य संबंध)

Definition: (a,b),(b,a)∈R⇒a=b(a,b), (b,a) \in R \Rightarrow a = b
Example: {(1,1), (2,2), (1,2)}
{(1,2), (2,1)} → Not anti-symmetric

4. Transitive Relation (सांक्रामक संबंध)

Definition: (a,b),(b,c)∈R⇒(a,c)∈R(a,b), (b,c) \in R \Rightarrow (a,c) \in R
Example: {(1,2), (2,3), (1,3)}

5. Equivalence Relation (समानता संबंध)

वह relation जो तीनों properties satisfy करता है:

Reflexive
Symmetric
Transitive

Example: “is equal to” (=) relation

6. Irreflexive Relation (अप्रत्यावर्ती संबंध)

(a,a)∉R(a,a) \notin R ∀ a ∈ A
Example: A = {1,2}, R = {(1,2), (2,1)} — Irreflexive

Visualization (Venn Diagram Style Explanation)

You can visualize relation as arrows from elements to elements.
Symmetric → double-sided arrow
Transitive → chains
Reflexive → self-loops

GATE और CS में उपयोग:

Concept Use Area
Reflexive Set Theory, Database Relations
Symmetric Undirected Graphs
Transitive Reachability in Graphs
Anti-Symmetric Partial Order Relations
Equivalence Classification, State Machines

Practice Example:

Let A = {1, 2, 3}

R = {(1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2)}

Q: यह relation कौन-कौन सी property satisfy करता है?

Reflexive?
Symmetric?
Transitive?
So, Not an Equivalence Relation

Extra Tip:

Equivalence Relation से बनता है:

Partition of Set — i.e., it divides the set into disjoint equivalence classes.

Conclusion:

Relation Type Symbolic Rule
Reflexive (a,a) ∈ R
Symmetric (a,b) ∈ R ⇒ (b,a) ∈ R
Anti-Symmetric (a,b) ∈ R ∧ (b,a) ∈ R ⇒ a = b
Transitive (a,b), (b,c) ∈ R ⇒ (a,c) ∈ R
Equivalence Reflexive + Symmetric + Transitive

क्या आप इस टॉपिक पर PDF notes, video lecture (Hindi), या MCQ practice sheet चाहते हैं?

Day 02-Discrete mathematics for computer science in Hindi – Type of Relation with basic concept

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