General Mathematics: Sets and Numbering Systems

Vocabulary practice flashcards covering fundamental set theory definitions, properties, numbering systems, and complex number operations.

Set

An unordered collection of distinct objects.

Element

An individual object contained in a set.

Roster Method

A way of specifying a set by listing its elements surrounded by braces (for example, $\{1, 2, 3, 4, 5\}$).

Set Builder Notation

A method of defining a set with the general format $\{\text{variable} \mid \text{descriptive statement}\}$, where the vertical bar is read as 'such that'.

Empty Set (Null Set)

The set containing no elements, designated by the symbol $\emptyset$.

Universal Set

The set of all objects or elements pertinent to a given discussion, designated by the symbol $U$.

Subset ($A \subseteq B$)

A relationship between two sets where every element of set $A$ is also an element of set $B$.

Proper Subset ($A \subset B$)

A relationship where set $A$ is a subset of set $B$, and $B$ contains at least one element that is not in $A$.

Equal Sets

Two sets $A$ and $B$ that contain the same elements, satisfying both $A \subseteq B$ and $B \subseteq A$, written as $A = B$.

Intersection ($A \cap B$)

The operation that produces a set containing only the elements that belong to both set $A$ and set $B$.

Union ($A \cup B$)

The operation that produces a set containing all elements that belong to set $A$ or set $B$.

Commutative Operations of Sets

The algebraic property stating that the order of sets does not change the result: $A \cup B = B \cup A$ and $A \cap B = B \cap A$.

Associative Operations of Sets

The algebraic property stating that grouping does not change the outcome: $(A \cup B) \cup C = A \cup (B \cup C)$ and $(A \cap B) \cap C = B \cap (A \cap C)$.

Distributive Property of Sets

The property stating that union and intersection distribute over each other: $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$ and $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$.

Natural Numbers ($\mathbb{N}$)

The counting numbers, denoted as $\mathbb{N} = \{1, 2, 3, \dots\}$.

Integers ($\mathbb{Z}$)

The set comprising natural numbers, their opposites (negatives), and zero, denoted as $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

Rational Numbers ($\mathbb{Q}$)

Numbers that can be expressed as a ratio of two integers ($\frac{a}{b}$ where $b \neq 0$), including fractions, terminating decimals, and repeating decimals.

Irrational Numbers

Numbers that cannot be written as fractions or ratios of integers, such as $\sqrt{2}$, $\pi$, and $e$.

Real Numbers ($\mathbb{R}$)

The complete set formed by the union of all rational numbers and irrational numbers, satisfying $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$.

Cardinality ($|A|$)

The number of elements contained in a set $A$.

Complex Numbers ($\mathbb{C}$)

Numbers of the form $a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i$ is the imaginary unit satisfying $i^2 = -1$.

Graphical Representation of a Complex Number

Representation of $a + bi$ in the complex plane as a point $p(a, b)$, where the $x$-axis represents real numbers and the $y$-axis represents imaginary numbers.

Complex Conjugate ($z^*$)

For a complex number $z = a + bi$, the value defined as $z^* = a - bi$.

Modulus (Absolute Value of a Complex Number)

The distance a complex number $z = a + bi$ is from the origin on the complex plane, calculated as $|z| = \sqrt{a^2 + b^2}$.