Relaciones y Funciones Matemáticas con Ejemplos

Concept of Function and Relation

Introduction to Functions and Relations

• The video begins by introducing the concept of function application or mapping, emphasizing the need to understand relations first.

• A relation is defined as a correspondence between two sets, referred to as set X (the first set) and set Y (the arrival set).

Understanding Relations

• A relation can be represented using ordered pairs; for example, if set M contains elements 5 and 7, and set N contains 3, 6, and 8.

• The combinations of these sets are illustrated through ordered pairs: (5,3), (5,6), (5,8), (7,3), (7,6), and (7,8).

• This relationship can also be visually represented with diagrams showing how elements from one set relate to another.

Transitioning to Functions

• After establishing what a relation is, the discussion shifts to functions—defined as a specific type of relation where each element in the first set corresponds uniquely to an element in the second.

Examples of Functions

• Three examples illustrate functions:

• First example relates numbers 1 through 4 with letters a through d.

• Second example uses colors associated with dogs demonstrating that multiple inputs can map to the same output without violating function rules.

Criteria for Valid Functions

• It’s clarified that not all relations qualify as functions. For instance:

• If an input has multiple outputs or if some inputs lack corresponding outputs in the second set.

Identifying Valid Functions

Analyzing Input Sets

• To determine if a relation is a function:

• Ensure every input has exactly one output.

• Check that no input maps to more than one output.

Examples of Non-functions

• Specific cases are analyzed where certain inputs have multiple outputs or lack any corresponding outputs altogether. These do not meet function criteria.

Confirming Valid Functions

• In contrast, valid functions are confirmed when:

• Each input has only one unique output even if different inputs share the same output.

Using Ordered Pairs and Value Tables

Final Examples with Tables

• The video concludes by presenting tables illustrating valid functions:

Understanding Ordered Pairs and Functions

Introduction to Ordered Pairs

• The discussion begins with an example of ordered pairs, highlighting the option to represent them either through a value table or a diagram.

• A diagram is chosen for illustration, where the first set is labeled as 'x' and the second set as 'y'.

Input Values

• The input values are identified from the left side of the diagram, which include: 1, 2 (noting that 1 appears again but will not be repeated), 4, and 5.

Output Values

• The output values corresponding to 'y' are listed as: 4, 6, 8, and 10.

Relationships Between Sets

• Connections between input and output values are established:

• The number 1 connects to both outputs: 2 and later again to 6.

• Other connections include:

• 2 with 4,

• 4 with 8,

• and finally,

• 5 with 10.

Conclusion on Functionality