# Sets

A **set** is a collection of different things. The things contained in a set are called **elements** or **members**. To denote the membership of \(a\) to a set \(A\) we write \(a\in A\) and read “a belongs in A” or “a is in A”. If we want to indicate that \(a\) is not a member of \(A\), we write \(a\not\in A\) and read “a does not belong in A” or “a is not in A”. A set without elements is called the empty set, and it is denoted by \(\emptyset\).

# Functions

A **function** is a rule that maps each element of a set \(X\) to exactly one element of a set \(Y\). The set \(X\) is called the **domain** of the function, and the set \(Y\) is called the **codomain** of the function. A function that maps \(X\) to \(Y\) is usually denoted by \(f\colon X \to Y\). For each element \(x\in X\), we write \(f(x)\) and read “f of x” to denote the element of \(Y\) to which \(x\) is mapped. A function with codomain the set of real numbers \(\mathbb{R}\) is called a **real-valued function**. A function having the set of real numbers as its domain is called a **function of a real variable**. It is usual to omit these specializations whenever they are understood from context, and simply refer to functions \(f\colon \mathbb{R} \to \mathbb{R}\) as functions instead of real-valued functions of a real variable.

# Differentiation

A function’s rate of change conveys very useful information about the nature of the rule that associates the domain and the codomain of the function. Suppose, for example, that \(C\) is a function describing the cost of a production process. For each desired output quantity \(q\in \mathbb{R}_{\ge 0}\), \(C(q)\) gives the minimum production cost for \(q\) (we call such functions *cost functions* in economics). Knowledge of \(C\) allows one to find the minimum cost for producing an output quantity of, say, \(5\). What if we want to calculate the incremental cost of “slightly increasing production” by a small amount? We can calculate the additional cost by examining the rate of change of the cost function (this is known in economics as the *marginal cost function*).

# Optimization

Decision problems in economics are predominantly described as optimization problems. For example, suppose that one has to decide among a finite number of alternatives \(\alpha_{1}, \alpha_{2}, \dots, \alpha_{n}\). Alternatives are evaluated based on a payoff function \(u\). The payoff she receives by choosing alternative \(\alpha_{j}\) is given by \(u(\alpha_{j})\). One way to model the agent’s decision is to assume that she chooses the alternative that maximizes the payoff she gains.

# A Hitchhiker's Guide to Mathematics for Economic Courses

We can only cover so much in a short preparatory course. This topic proposes a guide to help you navigate through your future economic studies.