Relations and Functions
Pichle chapter me humne sets seekhe. Ab dekhenge ki do sets ke elements aapas me kaise relate hote hain, aur ek special tarah ka relation jise function kehte hain — jo poore higher maths ki reedh ki haddi hai.
1. Ordered Pair aur Cartesian Product
Ek ordered pair \((a, b)\) me order matter karta hai: \((a,b) \neq (b,a)\) jab tak \(a \neq b\). Do ordered pairs equal tabhi jab dono components equal hon.
Sets \(A\) aur \(B\) ka Cartesian product \(A \times B\) saare ordered pairs ka set hai jahan pehla element \(A\) se aur doosra \(B\) se aaye: \[ A \times B = \{\, (a, b) : a \in A,\ b \in B \,\}. \]
Agar \(n(A) = p\) aur \(n(B) = q\) ho, to:
\[ n(A \times B) = p \times q. \]
\(A = \{1, 2\}\), \(B = \{x, y\}\). Toh \[ A \times B = \{(1,x),\,(1,y),\,(2,x),\,(2,y)\}, \qquad n(A\times B) = 2 \times 2 = 4. \]
2. Relation
Set \(A\) se set \(B\) tak ek relation \(R\), \(A \times B\) ka koi bhi subset hota hai. Agar \((a,b) \in R\) to likhte hain \(a\,R\,b\) ("\(a\) is related to \(b\)").
- Domain: \(R\) ke saare pehle elements (\(a\)) ka set.
- Range: \(R\) ke saare doosre elements (\(b\)) ka set.
- Codomain: poora set \(B\). Hamesha: range \(\subseteq\) codomain.
Agar \(n(A) = p\) aur \(n(B) = q\), to \(A \times B\) me \(pq\) elements hain, isliye \(A\) se \(B\) tak total relations \(= 2^{pq}\) (kyunki har subset ek relation hai).
\(A = \{1,2,3,4\}\) par \(R = \{(a,b) : b = a + 1\}\). Toh \[ R = \{(1,2),(2,3),(3,4)\}, \quad \text{Domain} = \{1,2,3\}, \quad \text{Range} = \{2,3,4\}. \]
3. Function
\(A\) se \(B\) tak ek function \(f\) ek aisa relation hai jisme \(A\) ke har element ka exactly ek image \(B\) me ho. Likhte hain \(f : A \to B\), aur \(b = f(a)\).
Yaad rakhne wali baat: function me domain ka har element use hona chahiye, aur kisi bhi element ke do images nahi ho sakte. (Ek image ko do alag elements share kar sakte hain — wo allowed hai.)
- \(f(a)\) = \(a\) ka image; \(a\) = \(f(a)\) ka pre-image.
- Domain = \(A\), Codomain = \(B\), Range = saare images ka set \(\subseteq B\).
4. Real-Valued Functions (Common Types)
Jab domain aur range dono real numbers ke subsets hon, to "real function" kehte hain. Kuch important:
| Function | Rule | Domain | Range |
|---|---|---|---|
| Identity | \(f(x) = x\) | \(\mathbb{R}\) | \(\mathbb{R}\) |
| Constant | \(f(x) = c\) | \(\mathbb{R}\) | \(\{c\}\) |
| Modulus | \(f(x) = |x|\) | \(\mathbb{R}\) | \([0, \infty)\) |
| Signum | \(f(x) = \dfrac{|x|}{x}\) (\(x\neq 0\)) | \(\mathbb{R}\) | \(\{-1, 0, 1\}\) |
| Greatest integer | \(f(x) = [x]\) | \(\mathbb{R}\) | \(\mathbb{Z}\) |
Modulus function:
\[ |x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases} \]
Signum function:
\[ \operatorname{sgn}(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0 \end{cases} \]
Greatest integer function \([x]\) = \(x\) se chhota ya barabar sabse bada integer. Jaise \([2.7] = 2\), \([-1.2] = -2\), \([3] = 3\).
5. Polynomial aur Rational Functions
- Polynomial: \(f(x) = a_0 + a_1 x + a_2 x^{2} + \dots + a_n x^{n}\). Domain \(= \mathbb{R}\).
- Rational: \(f(x) = \dfrac{p(x)}{q(x)}\), jahan \(p, q\) polynomials hain aur \(q(x) \neq 0\). Domain me \(q(x) = 0\) wale points hata dete hain.
6. Algebra of Real Functions
Do real functions \(f\) aur \(g\) ke liye (jahan dono defined hon):
- Sum: \((f + g)(x) = f(x) + g(x)\)
- Difference: \((f - g)(x) = f(x) - g(x)\)
- Product: \((fg)(x) = f(x)\,g(x)\)
- Scalar multiple: \((c f)(x) = c\,f(x)\)
- Quotient: \(\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}\), jahan \(g(x) \neq 0\).
\(f(x) = x^{2}\) aur \(g(x) = 2x + 1\). Toh \[ (f+g)(x) = x^{2} + 2x + 1, \qquad (fg)(x) = x^{2}(2x+1) = 2x^{3} + x^{2}. \]
Key Takeaways
- \(A \times B\) = saare ordered pairs ka set; \(n(A \times B) = n(A)\,n(B)\).
- Relation = \(A \times B\) ka subset; uska domain, range, codomain hota hai.
- Function = aisa relation jahan domain ke har element ka exactly ek image ho.
- Important real functions: identity, constant, modulus, signum, greatest integer, polynomial, rational.
- Functions ko add, subtract, multiply, divide kar sakte hain (domain ka dhyan rakhte hue).