Relations and Functions

Learning Objectives

• Understand the concept of the Cartesian product of sets, including ordered pairs, equality of ordered pairs, and the properties of Cartesian products such as $A \times B$, $A \times \emptyset = \emptyset$, $A \times B \neq B \times A$, and the cardinality $n(A \times B) = n(A)n(B)$. Interpret $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R} \times \mathbb{R} \times \mathbb{R}$.
• Define and represent relations as subsets of $A \times B$ using roster, set-builder, and arrow diagram forms. Identify the domain, range, codomain, image, and calculate the number of possible relations as $2^{pq}$.
• Characterize functions as a special type of relation where each element in the domain is associated with exactly one element in the codomain. Represent functions as mappings and identify their domain, codomain, and range.
• Perform operations on functions including addition, subtraction, multiplication, and division, defined pointwise for functions sharing the same domain.
• Analyze polynomial functions defined by expressions of the form $a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$, where $n$ is a non-negative integer and coefficients are real numbers.
• Examine rational functions as ratios of two polynomial functions, ensuring the denominator is non-zero.
• Explore the modulus function, which outputs the absolute value of a number, and the signum function, which indicates the sign of a number with outputs of -1, 0, or 1.
• Interpret the greatest integer function, denoted $[x]$, which returns the greatest integer less than or equal to $x$, resulting in a step-like graph.
• Determine the domain and range of real-valued functions from formulas, tables, ordered pairs, and graphs, including restrictions from square roots and denominators.
• Evaluate whether a relation is a function using ordered pairs, arrow diagrams, domain coverage, and the "one and only one image" condition.

Chapter Notes

Cartesian Product of Sets

• Ordered Pairs: An ordered pair $(a, b)$ is a pair of elements grouped together in a specific order.
• Equality of Ordered Pairs: Two ordered pairs $(a, b)$ and $(x, y)$ are equal if and only if $a = x$ and $b = y$.
• Cartesian Product: The Cartesian product $A \times B$ of two sets $A$ and $B$ is the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.
  • If either $A$ or $B$ is empty, $A \times B = \emptyset$.
  • Generally, $A \times B \neq B \times A$.
  • Cardinality: If $n(A) = p$ and $n(B) = q$, then $n(A \times B) = pq$.
• Examples:
  • $R \times R$ represents the set of all points in a two-dimensional space.
  • $R \times R \times R$ represents the set of all points in a three-dimensional space.

Relations

• Definition: A relation $R$ from a set $A$ to a set $B$ is a subset of the Cartesian product $A \times B$.
• Domain: The set of all first elements of the ordered pairs in a relation $R$.
• Range: The set of all second elements in a relation $R$.
• Codomain: The set $B$ in a relation $R$ from $A$ to $B$.
• Number of Relations: If $n(A) = p$ and $n(B) = q$, then the number of relations is $2^{pq}$.

Functions

• Definition: A function $f$ from a set $A$ to a set $B$ is a relation where each element of $A$ is associated with exactly one element in $B$.
• Properties:
  • Domain: The set $A$.
  • Codomain: The set $B$.
  • Range: The set of images of elements of $A$.
• Real-Valued Functions: Functions where the domain and range are subsets of real numbers.

Algebra of Functions

• Addition: $(f + g)(x) = f(x) + g(x)$.
• Subtraction: $(f - g)(x) = f(x) - g(x)$.
• Multiplication: $(fg)(x) = f(x)g(x)$.
• Scalar Multiplication: $(kf)(x) = kf(x)$, where $k$ is a real number.
• Division: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$, provided $g(x) \neq 0$.

Polynomial Functions

• Definition: A function of the form $f(x) = a_0 + a_1x + a_2x^2 + \ldots + a_nx^n$, where $n$ is a non-negative integer and coefficients are real numbers.

Rational Functions

• Definition: A function of the form $\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are polynomial functions and $g(x) \neq 0$.

Modulus and Signum Functions

• Modulus Function: $f(x) = |x|$, where $f(x) = x$ if $x \geq 0$ and $f(x) = -x$ if $x < 0$.
• Signum Function: $f(x) = \begin{cases} 1, & x > 0 \\ 0, & x = 0 \\ -1, & x < 0 \end{cases}$.

Greatest Integer Function

• Definition: $f(x) = [x]$, the greatest integer less than or equal to $x$.

Domain and Range of Real Functions

• Finding Domain and Range: Analyze the function's formula, considering restrictions such as square roots and denominators.

Function Testing and Mapping Diagrams

• Testing Relations for Functions: Use ordered pairs and arrow diagrams to ensure each element in the domain maps to exactly one element in the codomain.