8 Background: Functions
A function from a set
Definition 8.1 (Function) A function
Suppose that
. We write , and . We write , and . We write and
An example of a relation that is not a function is
When using functions, we often employ the following terminology and notation:
Definition 8.2 (Domain and Codomain) Suppose
Definition 8.3 (Image and Range) Suppose
For example, suppose that
- The domain of
is - The codomain of
is - The range of
is - The image of
is
In many situations, we need to apply one function to the output of another function. More formally, when the codomain of one function is the same as the domain of another function, we can compose these functions to create a new function:
Definition 8.4 (Composition of functions) Suppose that
For example, suppose
, , , ,
8.1 Exercises
Suppose that
and with , , .- Consider the function
defined by . What is ? - Consider the function
defined by . What is ? - Consider the function
defined by . What is ?
- Consider the function
In rational choice theory, we will often consider functions with domain and/or codomains that are powersets of a set. For example, suppose that
. A function from non-empty subsets of to non-empty subsets of is denoted . An example of such a function is: Does this function satisfy the following constraint: for all , ? If so, explain why. If not, explain why it fails the constraint and find a function that satisfies the constraint.
Suppose that
and with , , .Consider the function
defined by . What is ?Consider the function
defined by . What is ?Consider the function
defined by . What is ?
In rational choice theory, we will often consider functions with domain and/or codomains that are powersets of a set. For example, suppose that
. A function from non-empty subsets of to non-empty subsets of is denoted . An example of such a function is: Does this function satisfy the following constraint: for all , ? If so, explain why. If not, explain why it fails the constraint and find a function that satisfies the constraint.The above function does not satisfy this constraint since, for instance,
(we also have that ). An example of a function that satisfies the above constraint is: