Relations and Functions — Revision Notes
Punjab Board · Class 12 · Mathematics
Relations and Functions revision notes for Punjab Board Class 12 Mathematics: 4 topics in quick points. Includes Types of Relations, Types of Functions.
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Key Topics to Revise
Types of Relations
- A relation R in a set A is a subset of A × A (the Cartesian product of A with itself).
- Empty Relation: R = φ (no element is related to any other element). Example: R = {(a,b): a – b = 10} in A = {1,2,3,4} is empty.
- Universal Relation: R = A × A (every element is related to every element). Example: R = {(a,b): |a – b| ≥ 0} in A = {1,2,3,4} is universal.
Types of Functions
- One-One (Injective) Function: Different inputs always give different outputs. f(x₁) = f(x₂) ⟹ x₁ = x₂.
- Many-One Function: Two or more different inputs can give the same output. Example: f(x) = x² is many-one since f(1) = f(–1) = 1.
- Onto (Surjective) Function: Every element of the co-domain Y is the image of at least one element in the domain X. Range of f = Y.
Composition of Functions
- If f: A → B and g: B → C, then the composition gof: A → C is defined as gof(x) = g(f(x)) for all x ∈ A.
- The composition 'gof' means 'first apply f, then apply g'. Read from right to left.
- gof ≠ fog in general — composition is NOT commutative.
Invertible Functions
- A function f: X → Y is invertible if there exists a function g: Y → X such that gof = Iₓ and fog = Iy.
- g is called the inverse of f and is written as f⁻¹.
- CRITICAL: A function is invertible if and only if it is BIJECTIVE (both one-one and onto).
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