Sets

- 4 minutes read - 644 words

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 \(\emptyset\).

  • 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, \[ \times_{i=1}^{n} X_{i} = \left\{(x_{1}, \dots, x_{n})\colon\, x_{1}\in X_{1},\, \dots,\, x_{n}\in X_{n}\right\}. \]

Cartesian products are frequently used in economics and finance. For example, consider a situation where one would like to choose the amount of money invested in stocks and bonds. For simplicity, let \(X_{1} = \mathbb{R}_{\ge 0}\) be the set of potential stock investments in Euros, and \(X_{2} = \mathbb{R}_{\ge 0}\) be the set of potential bond investments in Euros (short sales are excluded). The Cartesian product \(X_{1}\times X_{2}\) is known as the 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 \(\alpha x_{1} + (1 - \alpha) x_{2} \in X\).

Topic's Concepts