Discrete Mathematics - Sets and Counting Vocabulary

Flashcard study deck covering core vocabulary, notation, types of sets, set operations, and number system classifications from Chapter 1.

Set

A collection of elements, or objects.

Tabular Notation

A method of representing a set by explicitly listing its members inside braces, presented as $x, y, \frac{}{}$ and read as 'the set whose members are $\{x, y, \frac{}{}\}$'.

Set-builder Notation

A method of representing a set by stating the properties or rules (a formula) that its members must satisfy, written generally as $\{x : x \text{ satisfies some rule}\}$.

Cardinality of a Set

The number of elements contained within a set, denoted for a set $A$ by $|A|$.

Subset

A collection of members of a set $A$ that are also members of a set $B$, denoted by $A \subseteq B$.

Proper Subset

A set $A$ that is a subset of set $B$ where $A$ is not equal to $B$, denoted by $A \subset B$.

Equal Sets

Two sets $A$ and $B$ that have the exact same members regardless of order, defined mathematically if and only if $A \subseteq B$ and $B \subseteq A$.

Equivalent Sets

Two sets $A$ and $B$ that have the same number of elements or equal cardinality ($|A| = |B|$), denoted by $A \equiv B$.

Universal Set

A set (denoted by $\xi$, $U$, or $S$) which contains all the elements or objects of other related sets without any repetition, representing all members under discussion in a given problem.

Empty Set

A set that contains no elements, also known as a null or void set, denoted by $\emptyset$ or $\{\}$.

Power Set

The set of all subsets of any set $A$, denoted by $P(A)$ or $2^A$.

Finite Set

A set having a finite or countable number of elements where the process of counting elements comes to an end.

Infinite Set

A set that is not finite, whose elements cannot be counted, or that has no last element.

Disjoint Sets

Two sets $A$ and $B$ that have no members in common, such that $A \cap B = \emptyset$.

Comparable Sets

Two sets $A$ and $B$ where one set is a subset of the other, meaning either $A \subseteq B$ or $B \subseteq A$.

Intersection of Sets

The collection of all members that belong to both set $A$ and set $B$, written in set-builder notation as $A \cap B = \{x : x \in A \text{ and } x \in B\}$.

Union of Sets

The collection of all members belonging to set $A$ or set $B$ (or both), written in set-builder notation as $A \cup B = \{x : x \in A \text{ or } x \in B \text{ or } x \in A \cap B\}$.

Difference of Sets

The collection of all members of set $A$ that are not in set $B$, denoted by $A - B$ or $A \setminus B = \{x : x \in A \text{ but } x \notin B\}$.

Complement of a Set

The set of elements not in set $A$ but present in the universal set, denoted by $A^c$ or $A'$ ($\xi - A$).

De Morgan's Laws

Set theory laws stating that $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$.

Natural Numbers

The set of positive integers used primarily for counting, denoted by $\mathbf{N} = \{1, 2, 3, \dots\}$.

Integers

The set of real whole numbers denoted by $\mathbf{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

Rational Numbers

Those real numbers which can be expressed as the ratio of two integers, defined as $\mathbf{Q} = \{x : x = \frac{p}{q}, \text{gcd}(p, q) = 1, \text{where } p, q \in \mathbf{Z}\}$.

Irrational Numbers

Those real numbers which are not rational, forming the complement of the set of rational numbers $\mathbf{Q}'$ in the real numbers $\mathbf{R}$.

Real Numbers

Numbers that can be paired off uniquely with points on a straight real line, denoted by $\mathbf{R}$.

Complex Numbers

Numbers of the form $a + ib$ where $a, b \in \mathbf{R}$, forming a superset of the real numbers ($\mathbf{R} \subset \mathbf{C}$).

Closed Set under an Operation

A property of a set of numbers where performing a mathematical operation on any two elements in the set yields an answer that is also contained in that set.