General Mathematics Lecture #1: Relations & Functions

Key Concepts in Relations & Functions

• A relation is defined as a set of ordered pairs (x, y).
• Example of a relation: A = {(1, 3), (2, 5), (3, 9), (4, 11), (1, −15)}.
• Ordered pairs are also known as coordinates, where:
  • Abscissa refers to the x-value.
  • Ordinate refers to the y-value.

Domain and Range

• Domain: The set of all x-values in a relation.
  • Example: Domain of set A = {1, 2, 3, 4}.
• Range: The set of all y-values in a relation.
  • Example: Range of set A = {3, 5, 9, 11, −15}.

Determining Domain and Range

• Various relations provided for practice:
  • A = {(−3, −4), (−2, 5), (−1, 5), (0, 6)}.
  • B = {(−2, 5), (0, 5), (2, 5), (4, 5)}.
  • C = {(0, 3), (0, 6), (−2, −4), (−4, −5)}.
  • D = {(1, 0), (0, 1), (−1, 0), (0, −1)}.
  • E = {(-4, 2), (-1, 3), (-4, 5), (-1, 0)}.

Understanding Functions

• A function is a special type of relation where no two distinct ordered pairs have the same first element (x-value).
• All functions are relations, but not all relations qualify as functions.

Ways to Represent a Function

• Numerically:
  • No shared x-coordinates among ordered pairs.
  • Representation through ordered pairs or a table of values.
• Graphically:
  • Use the vertical line test; a function's graph will intersect any vertical line at exactly one point.
• Mapping:
  • Types include one-to-one, one-to-many, many-to-one, and many-to-many correspondences.
• Algebraically:
  • The exponent of the y variable must not exceed one.
  • Example: Equation representation, such as 2x − 3y = 6.
• Verbally:
  • Formulated rules of correspondence expressed in words (like the relation between height and shoe size).

Function Verification

• A relation is a function if:
  • No two distinct pairs share the same x-coordinate.
  • A vertical line intersects the graph at one point.
  • In mappings, only one-to-one and many-to-one are considered functions.
  • Algebraic representation satisfies the degree condition of y.
  • Rules of correspondence indicate singularity in terms.

Classification of Relations and Functions

• Analyze given sets and equations to deem whether they are relations or functions.
• Examples provided for classification include:
  • A = {(1, 3), (2, 5), (3, 9), (4, 11), (1, −15)}.
  • Various mathematical relations and their contextual interpretations such as plate numbers or student references.
• Apply criteria to equations like y = √(x² − 1) or polynomials to assess if they define functions.

This quiz covers the fundamentals of relations and functions, focusing on defining key terms such as relation, domain, and range. Participants will determine the domain and range of various relations and assess whether given sets represent functions. It's an essential starting point for understanding mathematical relationships.