Cartesian Product of Sets, Relations & Functions
🔍 Introduction (परिचय):
Sets को एक साथ मिलाकर ordered pairs बनाना, उनके बीच relation दिखाना और function define करना — ये सभी चीज़ें Cartesian Product की मदद से संभव होती हैं।
इस भाग में हम सीखेंगे:
- Cartesian Product क्या है
- Relations कैसे बनते हैं
- Functions के प्रकार कौन-कौन से होते हैं
🔢 1. Cartesian Product of Sets
📚 Definition:
Let A और B दो sets हों।
तो उनका Cartesian Product A × B होता है: A×B={(a,b) ∣ a∈A and b∈B}
📌 इसका अर्थ: A से एक element और B से एक element का एक ordered pair बनता है।
🧠 Example:
Let A = {1, 2}, B = {x, y}
Then: A×B={(1,x),(1,y),(2,x),(2,y)}
📌 Total Pairs = 2 × 2 = 4
✅ Properties:
- A × B ≠ B × A (Generally)
क्योंकि ordered pairs में position मायने रखती है - If |A| = m, |B| = n ⇒ |A × B| = m × n
🔗 2. Relation (संबंध)
📚 Definition:
A relation R from set A to B is a subset of A × B.
यानि: R⊆A×B
📌 एक relation कुछ selected ordered pairs को represent करता है।
🧠 Example:
A = {1, 2, 3}, B = {4, 5}
Let relation R be: “a is less than b”
Then R = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5)}
But if we define relation R = {(1,4), (2,5)} ⇒ यह एक specific relation है
📊 Types of Relations (A → A):
- Reflexive: (a, a) ∈ R ∀ a ∈ A
- Symmetric: (a, b) ∈ R ⇒ (b, a) ∈ R
- Transitive: (a, b), (b, c) ∈ R ⇒ (a, c) ∈ R
- Equivalence Relation: जब relation reflexive, symmetric और transitive तीनों हो
🧮 3. Function (समाप्रेष) or Mapping
📚 Definition:
A function f from A to B is a relation where every element of A has exactly one image in B.
f: A→B
📌 Each input has only one output.
🧠 Example:
A = {1, 2, 3}, B = {a, b}
f = {(1, a), (2, b), (3, a)}
✅ यह एक function है क्योंकि हर element of A की एक ही image है।
❌ Example of Not a Function:
f = {(1, a), (1, b)}
🚫 Invalid क्योंकि 1 की दो images हैं
🔢 Types of Functions:
| Function Type | Description |
|---|---|
| One-One (Injective) | अलग-अलग input के लिए अलग-अलग output |
| Onto (Surjective) | हर B का element किसी न किसी A से जुड़ा हो |
| Bijective | One-One + Onto दोनों हो |
🎯 Objectives Recap:
- Cartesian product से ordered pairs बनाना
- Relation को subset के रूप में समझना
- Relation के types: reflexive, symmetric, transitive
- Function की पहचान और उनके प्रकार जानना
- Input-output mapping करना सीखना
📝 Exercise (अभ्यास प्रश्न):
📌 Short Answer Questions:
- Define Cartesian Product with an example.
- What is a relation from A to B?
- Define function. How is it different from a relation?
📌 Long Answer / Numerical Questions:
- Let A = {1, 2}, B = {3, 4}
a. Find A × B
b. List all possible relations from A to B
c. Is R = {(1, 3), (2, 4)} a function? - Prove with example:
a. A function is a special type of relation
b. Every bijective function has an inverse - Let R = {(1,1), (2,2), (3,3)} on set A = {1,2,3}
Is R reflexive, symmetric, transitive? Justify.