Do sets ke beech ka connection — Relations. Har input ka exactly ek output — Functions. Ye dono maths ki rooh hain!
Students = {Ali, Bhavna, Chetan} aur Marks = {85, 72, 91}. Agar Ali ne 85, Bhavna ne 72, Chetan ne 91 marks liye — toh yeh ek relation hai do sets ke beech. Aur kyunki har student ka exactly ek hi marks score hai, yeh ek function bhi hai! Lekin agar Ali ne do subjects mein alag marks liye hote (say 85 aur 90), toh woh sirf relation hota, function nahi.
Cartesian Product A × B woh set hai jisme A ke har element ko B ke har element ke saath pair banaya jaata hai. Ise "ordered pairs" kehte hain kyunki order matter karta hai: (a,b) ≠ (b,a).
Agar n(A) = m aur n(B) = n, toh n(A × B) = m × n ordered pairs honge.
Note: B×A = {(a,1),(a,2),(b,1),(b,2),(c,1),(c,2)} — yeh alag set hai!
A se B mein ek Relation R, A×B ka koi bhi subset hota hai. Matlab hum A×B mein se kuch ordered pairs choose karte hain — woh pairs hi relation banata hai.
Ek relation ko teen tareekon se represent karte hain:
Relation: A = {1,2,3} → B = {1,4,9} where (a,b) means b = a²
R = {(1,1), (2,4), (3,9)} — yeh function bhi hai (har input ka exactly ek output)
Domain: Relation mein jo first elements (x-values) hain — jahan se arrows nikalte hain. Yeh "input" set hai.
Codomain: Woh pura set B jahan arrows ja sakte hain — chahe sab elements use ho ya nahi.
Range: B ke sirf woh elements jinhe actually use kiya gaya — jahan arrows actually pahunche. Range ⊆ Codomain hamesha!
15 ∈ Codomain hai lekin 15 ∉ Range — kyunki koi input 15 map nahi karta.
Function ek special relation hai jisme Domain ke har element ka Codomain mein exactly ek image hota hai. Do conditions zaroori hain:
Alag inputs ke alag outputs. Koi do elements ek hi image nahi share karte: f(a) = f(b) → a = b
Codomain ka har element kisi na kisi input ka image hai. Range = Codomain exactly.
Dono conditions satisfy ho — perfect pairing! Inverse function tabhi exist karta hai.
Do ya zyada inputs ka same output. One-one nahi hai. Example: f(x) = x² (f(2)=f(−2)=4)
| Function | Definition | Example / Graph |
|---|---|---|
| Identity | Har element apne aap se map hota hai: f(x) = x | f(1)=1, f(5)=5 (diagonal line) |
| Constant | Har input ka same output: f(x) = c | f(x) = 7 (horizontal line) |
| Modulus |x| | Positive value return karta hai: |x| = x if x≥0, −x if x<0 | |−5| = 5, |3| = 3 (V shape) |
| Signum | Sign batata hai: sgn(x) = 1, 0, or −1 | sgn(−3)=−1, sgn(0)=0, sgn(4)=1 |
| Greatest Integer ⌊x⌋ | x se chota ya barabar sabse bada integer | ⌊3.7⌋=3, ⌊−1.2⌋=−2 (step function) |
Note: f(1) = 0 ka matlab x=1, f ka zero (root) hai!
Do functions ko add, subtract, multiply ya divide karke naye functions banate hain:
Q1. A = {1,2}, B = {3,4,5}. A×B ke saare elements likho aur total count batao.
A×B = {(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)} — Total = 2×3 = 6
Q2. f(x) = 3x−2 ke liye prove karo ki yeh one-one hai.
Maan lo f(a)=f(b) → 3a−2=3b−2 → 3a=3b → a=b. Toh f one-one hai ✓
Q3. f(x) = |x−2| ke liye f(0), f(2), f(5) nikalo.
f(0)=|0−2|=2 | f(2)=|2−2|=0 | f(5)=|5−2|=3
Q4. ⌊−3.5⌋ aur ⌊7.9⌋ ki value nikalo.
⌊−3.5⌋ = −4 (−3.5 se chota ya barabar sabse bada integer = −4) | ⌊7.9⌋ = 7
Q5. f(x) = x² aur g(x) = 2x+1 ke liye (f+g)(x) aur (f·g)(2) nikalo.
(f+g)(x) = x²+2x+1 = (x+1)² | (f·g)(2) = f(2)·g(2) = 4×5 = 20