Set theory is the foundational branch of mathematics that studies collections of objects and the rules governing their membership, containment, and combination — every other area of mathematics, from algebra to topology to computer science, uses set-theoretic language as its grammar. Relations extend this foundation by formally capturing how elements of one set correspond to elements of another, encoding structures as diverse as divisibility, database foreign keys, and scheduling dependencies all within a single framework. A crucial insight: a relation is simply a subset of a Cartesian product, which means every property of a relation (reflexivity, transitivity, antisymmetry) is really a constraint on which ordered pairs are included — recognizing this makes the classification of equivalences, orders, and functions feel inevitable rather than arbitrary.
What This Cheat Sheet Covers
This topic spans 14 focused tables and 109 indexed concepts, 95 flashcards. Below is a complete table-by-table outline of this topic, spanning foundational concepts through advanced details.
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Table 1: Set Notation and Representation
Sets are collections of distinct objects called elements; the notation used to describe them is the grammar of all further set-theoretic reasoning. Roster and set-builder notation are the two primary ways to write a set, and the membership and subset symbols are the connective tissue of every proof.
| Notation | Example | Description | |
|---|---|---|---|
${1, 2, 3, 4}$ | • Lists elements explicitly between braces • order and repetition do not change the set | ||
${x \in \mathbb{Z} \mid x > 0}$ | • Defines a set by a property: variable, separator ($\mid$ or $:$), predicate • Read "the set of all $x$ in $\mathbb{Z}$ such that $x > 0$." | ||
$3 \in {1,2,3}$; $5 \notin {1,2,3}$ | • Asserts an element belongs to a set • $\notin$ denotes non-membership | ||
$\emptyset = {}$, $\lvert\emptyset\rvert = 0$ | • The unique set with no elements • a subset of every set | ||
${1,2} \subseteq {1,2,3}$ | • $A \subseteq B$: every element of $A$ is also in $B$ • allows $A = B$. | ||
${1,2} \subset {1,2,3}$ | • $A \subset B$: $A \subseteq B$ and $A \neq B$ • $A$ is strictly smaller | ||
${1,2,3} \supseteq {1,2}$ | • $A \supseteq B$: $A$ contains all elements of $B$ • the converse of $\subseteq$. | ||
$A = B \Leftrightarrow A \subseteq B \wedge B \subseteq A$ | Two sets are equal iff they have exactly the same elements — proved by mutual containment. | ||
$U = {1,2,\ldots,n}$ in a given context | • The reference set containing all elements under discussion • all other sets are subsets of $U$. |