Mathematics/Discrete Math/Relations and Functions

Recall from Discrete Math - Sets (basic set operations) that a Cartesian Product is a mapping of ordered pairs from sets A={a1, a2, ...} and B={b1, b2, ...} such that A x B={(a1, b1), (a1, b2), ..., (a2, b1), (a2, b2), ...}.

Using this fact, we can define some terms:

• Binary Relation - A subset of the cartesian product A x B for some arbitrary, non-zero sets A and B.
  • When an element a ∈ A is paired with an element b ∈ B through a binary relation, we can denote that they're related with a R b.
• Function - A binary relation such that every element in A corresponds to, at most, one element in B.
• Domain - Given all {(a, b) ∈ f | a ∈ A, b ∈ B}, the "domain" is the full set A.
• Codomain - Given all {(a, b) ∈ f | a ∈ A, b ∈ B}, the "codomain" is the full set B.
• Argument - Given all {(a, b) ∈ f | a ∈ A, b ∈ B}, each a can be considered an "argument".
• Value - Given all {(a, b) ∈ f | a ∈ A, b ∈ B}, each b can be considered a "value".
• Finite Sequence - A function that has at most n domain → codomain pairings.
  • In simpler terms, a function that has a limited/set number of a → b mappings.
• Infinite Sequence - A function that has a non-terminating/infinitely extending number of domain → codomain pairings.

Note that two functions are considered equivalent if they have the same domain and codomain.