# Sets

- 4 minutes read - 644 wordsA **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 \(\emptyset\).

- A collection of four natural numbers

\[ A = \left\{4, 2, 1, 3\right\} \]

- A collection of colors

\[ B = \left\{\text{blue, white, red}\right\} \]

- The set of natural numbers

\[ \mathbb{N} = \left\{1, 2, 3, \dots \right\} \]

- The set of integers

\[ \mathbb{Z} = \left\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots \right\} \]

- The set of rational numbers

\[ \mathbb{Q} = \left\{\frac{z}{n}\colon\, z\in\mathbb{Z}, \, n\in\mathbb{N} \right\} \]

- The set of real numbers

\[ \mathbb{R} = \text{…needs more concepts to be described.} \]

- The set of solutions of the parabola \(2x^{2}-6x + 4 = 0\)

\[ S = \left\{x\in \mathbb{R} \colon\, 2x^{2}-6x + 4 = 0 \right\} = \left\{1, 2\right\} \]

Inclusion operators can be defined for sets. If every element of a set \(A\) is a member of a set \(B\), we write \(A\subseteq B\) and say that “A is a subset of B”. We can also say that “B is a superset of A”. Two sets \(A\) and \(B\) are equal if they have exactly the same elements, i.e., \(A \subseteq B\) and \(B \subseteq A\). We then write \(A = B\).

## Cartesian Products

The **Cartesian product** of two sets \(A\) and \(B\) is a set with elements pairs combining an element of \(A\) and an element of \(B\). Specifically, we write
\[
A \times B = \left\{(a,b)\colon\, a\in A,\, b\in B\right\}.
\]
Elements of the Cartesian product are denoted by \((a,b)\in A\times B\). Cartesian products are analogously defined of any finite collection of \(n\) sets. For example,
\[
\times_{i=1}^{n} X_{i} = \left\{(x_{1}, \dots, x_{n})\colon\, x_{1}\in X_{1},\, \dots,\, x_{n}\in X_{n}\right\}.
\]

*investment opportunity set*in finance, i.e., the set containing all investment choices available to an economic entity.

## Convex Combinations and Convex Sets

It is rare for economic choices to be made in isolation. When a person visits a retail store, she purchases a variety of products instead of a single one on most occasions. In production settings, entrepreneurs combine labor, capital, and other production factors to produce the desired output. In addition, entrepreneurs have to consider not only a particular pair of labor and capital but rather how this pair compares to other feasible pairs of capital and labor they could employ in production. A commonly used way (for reasons going beyond this introduction’s scope) to mathematically describe such collections of choices is via their convex combinations. For any real number \(\alpha\in[0,1]\), and any two points \(x_{1}, x_{2} \in X\), we say that \(x = \alpha x_{1} + (1 - \alpha) x_{2}\) is a **convex combination** of \(x_{1}\) and \(x_{2}\).

We say that a set \(X\) is convex if it contains all the convex combinations of its elements. Namely, \(X\) is a **convex set** if for every \(\alpha\in[0,1]\), and every \(x_{1}, x_{2} \in X\), we have \(\alpha x_{1} + (1 - \alpha) x_{2} \in X\).