1.3: Functions

• 1.3.1: Basics

  A function is a map which sends each element of a given set to a specific element in some (other) given set.

• 1.3.2: Kinds of Functions

  Functions are called surjective if every member of the codomain is a value of the function. Functions which never map different inputs to the same outputs are called injective. Functions which are both injective and surjective are called bijective.

• 1.3.3: Functions as Relations

  A function which maps elements of $A$ to elements of $B$ obviously defines a relation between $A$ and $B$, namely the relation which holds between $x$ and $y$ iff $f(x) = y$. In fact, we might even—if we are interested in reducing the building blocks of mathematics for instance—identify the function $f$ with this relation, i.e., with a set of pairs. This then raises the question: which relations define functions in this way?

• 1.3.4: Inverses of Functions

  We think of functions as maps. An obvious question to ask about functions, then, is whether the mapping can be “reversed.”

• 1.3.5: Composition of Functions

  We can define a new function by composing two functions, $f$ and $g$, i.e., by first applying $f$ and then $g$.

• 1.3.6: Isomorphism

  An isomorphism is a bijection that preserves the structure of the sets it relates, where structure is a matter of the relationships that obtain between the elements of the sets.

• 1.3.7: Partial Functions

  It is sometimes useful to relax the definition of function so that it is not required that the output of the function is defined for all possible inputs. Such mappings are called partial functions.

• 1.3.8: Summary