Relations and Functions
Class 11 me humne relation aur function ka basic idea seekha. Ab in dono ko gehraai se dekhenge — relations ke types (reflexive, symmetric, transitive), functions ke types (one-one, onto), aur composition aur inverse functions.
1. Types of Relations
Set \(A\) par defined relation \(R\) (yaani \(R \subseteq A \times A\)) ke kuch important types:
- Reflexive: har \(a \in A\) ke liye \((a, a) \in R\).
- Symmetric: \((a, b) \in R \Rightarrow (b, a) \in R\).
- Transitive: \((a, b) \in R\) aur \((b, c) \in R \Rightarrow (a, c) \in R\).
Jo relation teeno — reflexive, symmetric aur transitive — ho, use equivalence relation kehte hain.
Integers par "\(a\) aur \(b\) ka antar even hai" wala relation equivalence relation hai: har number ka khud se antar 0 (even) → reflexive; \(a-b\) even to \(b-a\) bhi even → symmetric; \(a-b\) aur \(b-c\) even to \(a-c\) even → transitive.
2. Types of Functions
Function \(f : A \to B\) ke types:
- One-one (injective): alag inputs ke alag outputs. \(f(x_1) = f(x_2) \Rightarrow x_1 = x_2\).
- Onto (surjective): \(B\) ka har element kisi na kisi ka image ho (range \(= B\)).
- Bijective: jo one-one aur onto dono ho.
Sirf bijective functions ka inverse hota hai.
\(f : \mathbb{R} \to \mathbb{R}\), \(f(x) = 2x + 3\). One-one: \(2x_1 + 3 = 2x_2 + 3 \Rightarrow x_1 = x_2\) ✓. Onto: har \(y\) ke liye \(x = \frac{y-3}{2}\) milta hai ✓. Toh \(f\) bijective hai.
3. Composition of Functions
\(f : A \to B\) aur \(g : B \to C\) ho, to composition \(g \circ f : A \to C\) aise: \[ (g \circ f)(x) = g\big(f(x)\big). \]
Dhyan: aam taur par \(g \circ f \neq f \circ g\) (order matter karta hai).
\(f(x) = x + 1\), \(g(x) = x^{2}\). Toh \[ (g \circ f)(x) = g(x+1) = (x+1)^{2}, \qquad (f \circ g)(x) = f(x^{2}) = x^{2} + 1. \] Dono alag — order important.
4. Invertible Functions
\(f : A \to B\) invertible hai agar koi \(g : B \to A\) mile aise ki \(g \circ f = I_A\) aur \(f \circ g = I_B\). Tab \(g = f^{-1}\). Function invertible tabhi jab wo bijective ho.
\(f(x) = 2x + 3\) ka inverse: \(y = 2x + 3 \Rightarrow x = \frac{y - 3}{2}\), toh \(f^{-1}(x) = \dfrac{x - 3}{2}\).
Key Takeaways
- Relations: reflexive, symmetric, transitive; teeno ho to equivalence relation.
- Functions: one-one (injective), onto (surjective), bijective (dono).
- Composition \((g \circ f)(x) = g(f(x))\); generally \(g \circ f \neq f \circ g\).
- Sirf bijective function invertible hota hai; inverse nikaalne ke liye \(y\) ke terms me \(x\) solve karo.