| --- |
| source: ncert |
| topic: functions |
| class_level: class_12 |
| chapter: ch1 |
| difficulty: advanced |
| --- |
| |
| Class 12 | Ch1: Relations and Functions |
| TYPES OF RELATIONS: |
| Empty relation: no element of A related to any element. R=phi. |
| Universal relation: every element related to every element. R=A×A. |
| Reflexive: (a,a) in R for all a in A. |
| Symmetric: if (a,b) in R then (b,a) in R. |
| Transitive: if (a,b) and (b,c) in R then (a,c) in R. |
| Equivalence relation: reflexive + symmetric + transitive. |
| EQUIVALENCE CLASSES: set of all elements related to a given element. |
| TYPES OF FUNCTIONS: |
| One-one (injective): f(a)=f(b) implies a=b. No two inputs give same output. |
| Onto (surjective): range=codomain. Every element of codomain has pre-image. |
| Bijective: one-one AND onto. Inverse function exists. |
| COMPOSITION: (fog)(x)=f(g(x)). gof≠fog generally. |
| If f and g are one-one then fog is one-one. If f and g are onto then fog is onto. |
| INVERTIBLE FUNCTIONS: |
| f is invertible iff f is bijective. |
| (f^(-1) o f)(x)=x and (f o f^(-1))(x)=x. |
| BINARY OPERATIONS: * on set A: A×A→A. |
| Commutative: a*b=b*a. Associative: (a*b)*c=a*(b*c). |
| Identity element e: a*e=e*a=a. Inverse of a: a*a^(-1)=e. |
| SOLVED EXAMPLES: |
| Example 1: R={(a,b): |a-b| divisible by 5} on integers. Is it equivalence? |
| Reflexive: |a-a|=0 divisible by 5. Yes. |
| Symmetric: |a-b| div by 5 → |b-a|=|a-b| div by 5. Yes. |
| Transitive: |a-b|=5k, |b-c|=5m → |a-c|<=|a-b|+|b-c|=5(k+m). Yes. |
| Equivalence relation confirmed. |
| Example 2: f:R→R, f(x)=2x+3. Show bijective and find inverse. |
| One-one: f(a)=f(b) → 2a+3=2b+3 → a=b. Yes. |
| Onto: for any y, x=(y-3)/2 in R. Yes. Bijective. |
| f^(-1)(x)=(x-3)/2. |
| COMMON MISTAKES: |
| Equivalence class [a]={b in A: (a,b) in R}. Different elements may have same class. |
| A function must be bijective for inverse to exist (not just one-one or just onto). |
| fog means g applied FIRST then f. |
