Set Theory Fundamentals

Author

John Robin Inston

Published

October 11, 2026

1 Set Theory Fundamentals

This note collects the basic vocabulary and operations of naïve [[set-theory|set theory]] that are used throughout analysis and measure theory. De Morgan’s laws, which relate unions, intersections and complements, are included under Algebra of Sets below.

2 Sets and Subsets

A set is a collection of distinct objects, called its elements. We write \(x\in A\) if \(x\) is an element of \(A\). The empty set \(\emptyset\) has no elements. A set \(A\) is a subset of \(B\), written \(A\subseteq B\), if \(x\in A\implies x\in B\). Two sets are equal if and only if \(A\subseteq B\) and \(B\subseteq A\). The power set of \(X\) is the collection of all subsets of \(X\), \[ 2^X:=\{A: A\subseteq X\}. \]

Note: If \(X\) is finite with \(n\) elements then \(\lvert 2^X\rvert=2^n\), which motivates the notation \(2^X\).

Throughout, we fix an ambient universe set \(X\) and consider subsets of \(X\).

3 Set Operations

For \(A,B\subseteq X\) we define the

  • union \(A\cup B:=\{x: x\in A \text{ or } x\in B\}\);
  • intersection \(A\cap B:=\{x: x\in A \text{ and } x\in B\}\);
  • complement \(A^c:=\{x\in X: x\notin A\}\);
  • difference \(A\setminus B:=\{x: x\in A \text{ and } x\notin B\}=A\cap B^c\);
  • symmetric difference \(A\triangle B:=(A\setminus B)\cup(B\setminus A)\).

4 Indexed Families

Let \(\{A_\alpha\}_{\alpha\in I}\) be a family of subsets of \(X\) indexed by a set \(I\). Then \[ \bigcup_{\alpha\in I}A_\alpha:=\{x: x\in A_\alpha \text{ for some }\alpha\in I\}, \qquad \bigcap_{\alpha\in I}A_\alpha:=\{x: x\in A_\alpha \text{ for all }\alpha\in I\}. \]

When \(I=\mathbb{N}\) we have a countable union or intersection, which is the form that appears in the definition of a \(\sigma\)-algebra.

5 Algebra of Sets

For \(A,B,C\subseteq X\):

  1. Commutativity: \(A\cup B=B\cup A\) and \(A\cap B=B\cap A\).
  2. Associativity: \((A\cup B)\cup C=A\cup(B\cup C)\) and \((A\cap B)\cap C=A\cap(B\cap C)\).
  3. Distributivity: \(A\cap(B\cup C)=(A\cap B)\cup(A\cap C)\) and \(A\cup(B\cap C)=(A\cup B)\cap(A\cup C)\).
  4. Complements: \((A^c)^c=A\), \(A\cup A^c=X\), \(A\cap A^c=\emptyset\).
  5. De Morgan: \(\left(\bigcup_\alpha A_\alpha\right)^c=\bigcap_\alpha A_\alpha^c\) and \(\left(\bigcap_\alpha A_\alpha\right)^c=\bigcup_\alpha A_\alpha^c\).

Proof. Each identity is proved by double inclusion: take an arbitrary \(x\) in the left-hand side and show it lies in the right-hand side, then vice versa. For example, for the first distributive law, \[ x\in A\cap(B\cup C)\iff x\in A \text{ and } (x\in B \text{ or } x\in C) \iff (x\in A\cap B) \text{ or } (x\in A\cap C). \]

6 Disjoint Unions and Partitions

Sets \(A,B\) are disjoint if \(A\cap B=\emptyset\). A family \(\{A_\alpha\}_{\alpha\in I}\) is pairwise disjoint if \(A_\alpha\cap A_\beta=\emptyset\) for all \(\alpha\neq\beta\). We write \(A\sqcup B\) or \(\bigsqcup_\alpha A_\alpha\) for a union that is known to be disjoint. A partition of \(X\) is a pairwise disjoint family of nonempty sets whose union is \(X\).

Any union can be made disjoint: given \(\{A_n\}_{n\geq1}\), set \(B_1=A_1\) and \(B_n=A_n\setminus\bigcup_{k<n}A_k\). Then the \(B_n\) are pairwise disjoint and \(\bigcup_nB_n=\bigcup_nA_n\). This trick is used constantly to reduce statements about measures of countable unions to countable additivity.

7 Cartesian Products

The Cartesian product of sets \(A\) and \(B\) is the set of ordered pairs \[ A\times B:=\{(a,b): a\in A,\ b\in B\}. \] More generally, for an indexed family, \(\prod_{\alpha\in I}A_\alpha\) is the set of functions \(x\) on \(I\) with \(x(\alpha)\in A_\alpha\) for all \(\alpha\). We write \(A^n=A\times\cdots\times A\).

8 Monotone Sequences of Sets

A sequence \(\{A_n\}_{n\geq1}\) is increasing if \(A_n\subseteq A_{n+1}\) for all \(n\), in which case we write \(A_n\uparrow A\) with \(A=\bigcup_nA_n\). It is decreasing if \(A_{n+1}\subseteq A_n\) for all \(n\), in which case we write \(A_n\downarrow A\) with \(A=\bigcap_nA_n\).

Monotone sequences are the sets analogue of monotone real sequences, and they are the setting for continuity from below and above of measures.

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