This chapter introduces the fundamental concepts of set theory, which are essential for understanding Brownian motion, random variables, probability theory, information economics, and game theory. Set theory is not typically covered in formal economist training, so the chapter provides a detailed discussion of the necessary topics. A set is a collection of objects, specified by curly brackets. For example, the set \( M \) contains real numbers \( x \) such that \( ax + bx^2 \geq 0 \). Sets can be written more compactly, such as \( (0, 1) \) for the set of real numbers between 0 and 1. Common sets include \( \mathbb{R} \) (real numbers), \( \mathbb{N} \) (natural numbers), \( \mathbb{Z} \) (integers), and \( \mathbb{Q} \) (rational numbers). The empty set \( \emptyset \) contains no elements. Set operations include union, intersection, and difference. The union of two sets \( A \) and \( B \) is denoted by \( A \cup B \) and contains all elements from both sets. The intersection \( A \cap B \) contains elements common to both sets. disjoint sets have an empty intersection. The difference \( A \backslash B \) contains elements in \( A \) but not in \( B \). Set operations follow arithmetic rules, such as \( C \backslash (A \cap B) = (C \backslash A) \cup (C \backslash B) \) and \( C \backslash (A \cup B) = (C \backslash A) \cap (C \backslash B) \). Infinite sets can be unioned and intersected. The infinite union \( \bigcup_{n=1}^{\infty} A_n \) contains all elements from each set \( A_n \). The infinite intersection \( \bigcap_{n=1}^{\infty} A_n \) contains only elements present in all sets \( A_n \). For example, \( \mathbb{N} = \bigcup_{n=0}^{\infty} \{n\} \) and \( \emptyset = \bigcap_{n=0}^{\infty} [n, \infty) \). The power set \( \mathcal{P}(\Omega) \) is the set of all subsets of \( \Omega \). For a finite set with \( n \) elements, the power set has \( 2^n \) subsets. For example, \( \mathcal{P}(\{1, 2, 3\}) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1,This chapter introduces the fundamental concepts of set theory, which are essential for understanding Brownian motion, random variables, probability theory, information economics, and game theory. Set theory is not typically covered in formal economist training, so the chapter provides a detailed discussion of the necessary topics. A set is a collection of objects, specified by curly brackets. For example, the set \( M \) contains real numbers \( x \) such that \( ax + bx^2 \geq 0 \). Sets can be written more compactly, such as \( (0, 1) \) for the set of real numbers between 0 and 1. Common sets include \( \mathbb{R} \) (real numbers), \( \mathbb{N} \) (natural numbers), \( \mathbb{Z} \) (integers), and \( \mathbb{Q} \) (rational numbers). The empty set \( \emptyset \) contains no elements. Set operations include union, intersection, and difference. The union of two sets \( A \) and \( B \) is denoted by \( A \cup B \) and contains all elements from both sets. The intersection \( A \cap B \) contains elements common to both sets. disjoint sets have an empty intersection. The difference \( A \backslash B \) contains elements in \( A \) but not in \( B \). Set operations follow arithmetic rules, such as \( C \backslash (A \cap B) = (C \backslash A) \cup (C \backslash B) \) and \( C \backslash (A \cup B) = (C \backslash A) \cap (C \backslash B) \). Infinite sets can be unioned and intersected. The infinite union \( \bigcup_{n=1}^{\infty} A_n \) contains all elements from each set \( A_n \). The infinite intersection \( \bigcap_{n=1}^{\infty} A_n \) contains only elements present in all sets \( A_n \). For example, \( \mathbb{N} = \bigcup_{n=0}^{\infty} \{n\} \) and \( \emptyset = \bigcap_{n=0}^{\infty} [n, \infty) \). The power set \( \mathcal{P}(\Omega) \) is the set of all subsets of \( \Omega \). For a finite set with \( n \) elements, the power set has \( 2^n \) subsets. For example, \( \mathcal{P}(\{1, 2, 3\}) = \{\emptyset, \{1\}, \{2\}, \{3\}, \{1, 2\}, \{1,