Relations and Functions

• Cartesian Product
  • Definition of Ordered Pair
  • Set $A \times B = {(a, b) : a \in A, b \in B}$
  • Number of elements: $n(A \times B) = p \times q$
  • Equality of ordered pairs
• Relations
  • Definition: Subset of $A \times B$
  • Domain: Set of first elements
  • Range: Set of second elements
  • Codomain: Entire set $B$
  • Total relations from $A$ to $B$: $2^{n(A \times B)}$
  • Types of Relations (Class 12 prep)
    • Reflexive, Symmetric, Transitive
    • Equivalence Relation
• Functions
  • Definition: Special type of relation (Every $a \in A$ has a unique $b \in B$)
  • Notation: $f: A \to B$
  • Domain, Codomain, and Range of a function
  • Graph of a function
  • Real-valued functions
• Special Types of Functions
  • Identity Function: $f(x) = x$
  • Constant Function: $f(x) = c$
  • Polynomial Function: $f(x) = a_0 + a_1x + \dots + a_nx^n$
  • Rational Function: $\frac{f(x)}{g(x)}$ where $g(x) \neq 0$
  • Modulus Function: $f(x) = |x|$
  • Signum Function: $\text{sgn}(x)$
  • Greatest Integer Function: $f(x) = [x]$
• Algebra of Functions
  • Addition: $(f + g)(x) = f(x) + g(x)$
  • Subtraction: $(f - g)(x) = f(x) - g(x)$
  • Scalar Multiplication: $(cf)(x) = c \cdot f(x)$
  • Multiplication (Product): $(fg)(x) = f(x) \cdot g(x)$
  • Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}, g(x) \neq 0$