Summary of "Lec 07 - Functions"

Main Ideas and Concepts

Definition of Functions:
A function is a rule that converts an input (X) into an output (Y). Example: The function f(x) = x^2 maps input X to its square.
Notation:
Functions are often denoted as f: X → Y, where X is the domain (input set) and Y is the codomain (output set).
Domain, Codomain, and Range:
- Domain: The set of all possible inputs for the function.
- Codomain: The set of potential outputs, which may include values not actually produced by the function.
- Range: The actual set of outputs produced by the function, which is a subset of the codomain.
Graphical Representation:
Functions can be represented graphically, with the relation of inputs and outputs plotted on a graph.
Properties of Functions:
- Injective (One-to-One): Different inputs produce different outputs.
- Surjective (Onto): Every element in the codomain is an output of the function.
- Bijective: A function that is both Injective and surjective, establishing a one-to-one correspondence between the domain and codomain.
Examples of Functions:
- Linear Functions of the form mx + c produce straight lines, with slope (m) affecting the angle and intercept (c) affecting where the line crosses the y-axis.
- The Square root function is defined to produce only non-negative outputs, thus limiting its domain.
Cardinality and Bijection:
Establishing a Bijection is crucial for comparing the sizes (Cardinality) of sets, especially infinite sets. Example: The number of lines that can be drawn in a plane corresponds to the number of points in that plane, showing that both sets have the same Cardinality.

Methodology and Instructions

Defining a Function:
- Specify the rule (e.g., f(x) = x^2).
- Clearly define the domain and codomain.
- Identify the range of the function.
Determining Properties:
- To check if a function is Injective, verify that different inputs yield different outputs.
- To check if a function is surjective, ensure every element in the codomain is an output for some input.
- For Bijection, confirm that the function is both Injective and surjective.
Using Bijection for Cardinality:
To establish that two sets have the same Cardinality, demonstrate a Bijection between them.