Functions

• A function has three things
  • two sets, the domain and codomain
  • a rule that associates any element in the domain to exactly one element in the codomain
• It must possible to evaluate the function on every element of the domain, and every output must be in the codomain
  • However, it is not needed for every element in the codomain to be a value of the function
  • The set of element actually reached by the function is called the image

Definition: Image. The set of all values of is called its image: is an element of the image of a function if there exists an such that Example: The codomain of the squaring function given by is , while the image is the nonnegative real numbers. Always, the image will be the subset of the codomain. Definition: Preimage. The preimage of some function , denoted as , gives the set of the values that map to some set. Formally, the preimage of some set , is

• A function can also be described as a relation in where each element of is related to a unique element of . This can be described as
  • This definition is what was introduced in lecture
    • The relation set is denoted as
  • For one , there is only one order pair in the relation set.
• Functions can be injective, surjective, or both (bijective)