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.

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