Math 223 Gottlieb

Functions Done Right

A map f: X → Y is a rule f which assigns to every object x in the source X an object f(x) in the target Y.

Definition 2: Two sets X and Y are equal, denoted X = Y, ⇔ every object of X is an object in Y and every object in Y is an object in X.

Definition 3: Two maps are equal ⇔ Their sources are equal, their targets are equal, and their rules do the same thing.

In more symbolic notation this definition reads as follows: Let f: X → Y and g: X′ → Y′ be two maps . Then f = g ⇔ X = X′, and Y = Y′ and f(a) = g(a) for all a in the source.

Definition 4. Suppose we have two maps, f: X → Y and g: Y → Z (note the target of f is the source of g). Then the map h: X → Z is the composition of f and g ⇔ h(x) = g(f(x)) for every x in the source X.

We say that f(x) is the image of x, and that x is a preimage of f(x). The set of all the preimages of y we will call the fiber of y.

The subset of the target Y which consists of elements y assigned to some x is called the image of f. Thus we can say that the image of f is the subset of objects of the target with non–empty fibers.

If A is a subset of the source X, then the image of A in the target Y consists of the set of images of all the objects in A. The image of A is denoted by f(A). Similarly, the preimage of a subset B of the target Y is the set of all objects in the source X whose images are objects in B. The preimage of B is denoted by f^{−1}(B).

If the image of f is the entire target of f, then we say that f is onto. Thus f is onto if and only if no fiber is empty. If the fibers of f have at most one object in it, then we say f is one–to–one. If every fiber of f has exactly one object in it, then we say that f is one–to–one onto or equivalently that f is bijective.