Learning Objectives
- Understand ordered pairs and Cartesian products
- Learn about relations and their types
- Study functions, their types, and algebra of functions
Key Concepts
Cartesian Product
Ordered pair: (a, b) where order matters; (a, b) ≠ (b, a) unless a = b. Cartesian product: A × B = {(a, b) : a ∈ A, b ∈ B}. If |A| = m, |B| = n, then |A × B| = mn. A × B ≠ B × A (unless A = B or A = φ or B = φ). R × R = R2 (coordinate plane). A × B × C = {(a, b, c) : a ∈ A, b ∈ B, c ∈ C}.
Relations
A relation R from set A to set B is a subset of A × B. Written as aRb if (a, b) ∈ R. Domain: Set of all first elements. Range: Set of all second elements. Co-domain: Set B itself (range ⊆ co-domain). Number of relations from A (m elements) to B (n elements) = 2mn.
Functions
A function f: A → B is a relation where every element of A has exactly one image in B. A = domain, B = co-domain. f(x) denotes image of x. No element of A is left unrelated. No element of A has more than one image.
Types of functions: One-one (Injective): f(x₁) = f(x₂) implies x₁ = x₂ (distinct elements have distinct images). Onto (Surjective): Range = Co-domain (every element of B has a pre-image). Bijective: Both one-one and onto. Many-one: Two or more elements have the same image. Into: Range is a proper subset of co-domain.
Algebra of Functions
For functions f, g: X → R: (f + g)(x) = f(x) + g(x); (f - g)(x) = f(x) - g(x); (fg)(x) = f(x) · g(x); (f/g)(x) = f(x)/g(x), g(x) ≠ 0; (kf)(x) = k · f(x) for scalar k.
Important Functions and Their Graphs
Identity function: f(x) = x. Constant function: f(x) = c. Polynomial function: f(x) = a₀ + a₁x + ... + aₙxn. Modulus function: f(x) = |x| (V-shaped graph). Signum function: sgn(x) = 1 (x > 0), 0 (x = 0), -1 (x < 0). Greatest integer function: f(x) = [x] (step function). Fractional part function: {x} = x - [x].
Summary
Cartesian product of sets creates ordered pairs. Relations are subsets of Cartesian products. Functions are special relations where every domain element has exactly one image. Functions can be classified as one-one, onto, bijective, or many-one. Algebra of real-valued functions includes addition, subtraction, multiplication, and division.
Important Terms
- Ordered pair: (a, b) where sequence matters
- Cartesian product: A × B = set of all ordered pairs from A and B
- Domain: Set of all input values of a function
- Range: Set of all output values of a function
- Injective: One-one function; distinct inputs give distinct outputs
- Surjective: Onto function; range equals co-domain
- Bijective: Function that is both injective and surjective
- Greatest integer function: [x] = greatest integer less than or equal to x
Quick Revision
- |A × B| = |A| × |B|
- Number of relations from A to B: 2|A|×|B|
- Number of functions from A to B: |B||A|
- Function: every element of domain has exactly one image
- Injective: f(a) = f(b) ⟹ a = b
- |x| = x if x ≥ 0; |x| = -x if x < 0
- Greatest integer [3.7] = 3; [-2.3] = -3
- Fractional part: {x} = x - [x], always in [0, 1)