Set Theory: Notation and Forms

Vocabulary flashcards covering fundamental set theory concepts, set notation, special sets, set types, and ways to represent sets.

Set

A well-defined collection of objects, typically denoted by capital letters such as $A$, $B$, or $C$.

Element

An object that forms part of a set, typically enclosed inside braces $\{\}$.

Membership Symbol ($\in$)

A mathematical symbol read as "is an element of" or "belongs to". For example, given $C = \{1, 2, 3, 4, 5\}$, the statement $2 \in C$ is true.

Non-membership Symbol ($\notin$)

A mathematical symbol read as "does not belong to". For example, given $C = \{1, 2, 3, 4, 5\}$, the statement $6 \notin C$ is true.

Empty Set or Null Set ($\emptyset$ or $\{\}$)

A set that contains no elements, such as the set of people older than $150$ years.

Universal Set ($U$)

The set that contains all elements under consideration within a given context.

Finite Set

A set containing a specific, countable number of elements, such as $D = \{1, 2, 3, 4, \dots, 163\}$.

Infinite Set

A set for which it is impossible to list the totality of its elements, such as $\mathbb{N}$ or $\mathbb{Z}$.

Extensive Form (Roster Form)

A method of representing a set by explicitly listing all of its elements inside braces, such as $A = \{2, 4, 6, 8\}$.

Comprehensive Form (Set-Builder Notation)

A method of representing a set by specifying a condition or property $p$ that all its elements satisfy, written as $\{x \mid x \text{ has property } p\}$.

Set of Natural Numbers ($\mathbb{N}$)

An infinite set of positive counting numbers, represented as $\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$.

Set of Integers ($\mathbb{Z}$)

An infinite set containing positive whole numbers, zero, and negative whole numbers, represented as $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

Set-Builder Inequality Symbols

Symbols used in set-builder notation to express intervals: $>$ (greater than), $\ge$ (greater than or equal to), $<$ (less than), and $\le$ (less than or equal to).