Sets, Relations, Functions, and Binary Operations Vocabulary

Flashcards covering key mathematical terminology, standard number sets, set operations, relations, functions, and binary operations.

Set

A collection of well-defined, distinct objects called elements.

Natural Numbers ($\mathbb{N}$)

The set of counting numbers starting from $1$, defined as $\{1, 2, 3, \dots\}$.

Whole Numbers ($\mathbb{W}$)

The set of non-negative integers starting from $0$, defined as $\{0, 1, 2, 3, \dots\}$.

Integers ($\mathbb{Z}$)

The set of whole numbers and their negative opposites, defined as $\{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

Rational Numbers ($\mathbb{Q}$)

Numbers that can be written in the form $\frac{a}{b}$, where $a, b \in \mathbb{Z}$ and $b \neq 0$; represented in decimal form as terminating or non-terminating repeating decimals.

Irrational Numbers ($\mathbb{Q}^c$)

Real numbers that cannot be written as a ratio of two integers (when the universal set is $\mathbb{R}$, $\mathbb{Q}^c = \mathbb{R} - \mathbb{Q}$); represented by non-terminating, non-repeating decimals such as $\sqrt{2}$ and $\pi$.

Real Numbers ($\mathbb{R}$)

The set containing all rational and irrational numbers.

Roster / Listing Method

A method to describe a set by explicitly listing all of its elements inside curly braces (e.g., $A = \{2, 4, 6, 8\}$).

Set-Builder / Rule Method

A method to describe a set by stating the property that determines membership (e.g., $A = \{x \in \mathbb{N} \mid x \text{ is even and } x < 10\}$).

Subset ($\subseteq$)

A relation where $A \subseteq B$ means that every element of set $A$ is also an element of set $B$.

Universal Set ($U$)

A set containing all objects currently under consideration in a specific context.

Complement ($A^c$)

The set of elements in the universal set $U$ that are not contained in set $A$.

Union ($A \cup B$)

The set operation that yields all elements belonging to set $A$, set $B$, or both.

Intersection ($A \cap B$)

The set operation that yields only the elements common to both set $A$ and set $B$.

Difference ($A - B$)

The set operation that yields the elements present in set $A$ but not in set $B$.

Cartesian Product ($A \times B$)

The set of all ordered pairs $(a, b)$ such that $a \in A$ and $b \in B$.

Relation

Any selected set of ordered pairs from the Cartesian product $A \times B$ that connects objects or records.

Domain

The set of all first components (inputs) that appear in a relation.

Range

The set of all second components (outputs) that appear in a relation.

Function

A special type of relation in which every input is paired with exactly one output.

Vertical Line Test

A visual test for graphs where a curve represents a function if and only if every vertical line meets the graph at most once.

Binary Operation

An operation $\star$ on a set $A$ that takes two elements of $A$ and combines them under a rule to produce one element of the same set $A$ ($\star : A \times A \rightarrow A$).

Closure

The essential property of a binary operation stating that the output produced by combining two elements of a set must remain inside that same set.

Venn Diagram Relationships

Visual diagrams showing set relationships: disjoint sets share no common elements, overlapping sets share some elements, and a subset lies entirely inside another set.