Mathematical Language and Set Theory Lecture Notes

These flashcards cover vocabulary related to mathematical language, set theory fundamentals, and the basic concepts of relations and functions.

Mathematical Language

A system used to communicate mathematical ideas consisting of natural language, technical terms, grammatical conventions, and a highly specialized symbolic notation.

Characteristics of Mathematical Language

The three main characteristics: being precise, concise, and powerful.

Context

The particular topic being studied or the environment in which one is working.

Convention

Where mathematicians and scientists have decided that particular symbols will have particular meanings; a fact, name, notation, or usage generally agreed upon.

Expression

A finite combination of symbols used to represent an object of interest that does not contain a complete thought and cannot be determined to be true or false.

Sentence

A correct arrangement of mathematical symbols that states a complete thought and can be determined to be true, false, or sometimes true/sometimes false.

PEMDAS

A mathematical convention for the order of operations: Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction.

Georg Cantor

The German mathematician (1845–1918) who introduced the study of sets as a fundamental theory in Mathematics in the 1870s.

Set theory

The branch of Mathematics that studies sets or the mathematical science of the infinite.

Set

A well-defined collection of objects.

Elements

The objects that make up a set, also referred to as members.

$$\in$$

The symbol used to denote that an object is an element of a set.

$$\notin$$

The symbol used to denote that an object is not an element of a set.

Descriptive form

A way to represent a set by giving a verbal description of its elements.

Roster method

A way to represent a set where elements are enumerated and separated by commas; also known as the Tabulation method.

Rule method

A way to represent a set by describing its elements, also called Set builder notation.

$$|$$

In set builder notation, this symbol is read as 'such that'.

Finite set

A set whose elements are limited or countable, and the last element can be identified.

Infinite set

A set whose elements are unlimited or uncountable, and the last element cannot be specified.

Unit set

A set with only one element, also called a singleton.

Null set

A set with no elements, also called an empty set.

Cardinal number

The number of elements or members in a set, often denoted as $n(A)$.

Subset

Set $A$ is a subset of $B$ ($A \subseteq B$) if and only if every element of $A$ is also an element of $B$.

Proper subset

Set $A$ is a proper subset of $B$ ($A \subset B$) if every element of $A$ is in $B$ but there is at least one element of $B$ that is not in $A$.

Power set

The collection of all subsets of a given set.

Union

The set of all elements $x$ such that $x$ is in $A$ or $x$ is in $B$, denoted by $A \cup B$.

Intersection

The set of all elements $x$ such that $x$ is in $A$ and $x$ is in $B$, denoted by $A \cap B$.

Disjoint sets

Sets that have no elements in common, meaning their intersection is the null set ($A \cap B = \emptyset$).

Cartesian product

The set of all ordered pairs $(a, b)$ where $a$ is in $A$ and $b$ is in $B$, denoted $A \times B$ (read as 'A cross B').

Relation

A well-defined relationship between two sets of numbers, formally defined as a set of ordered pairs.

Function

A special kind of relation where every input ($x$-value) is associated with exactly one output ($y$-value).

Mapping diagram

A representation of relations and functions consisting of two parallel columns showing how inputs (domain) correspond to outputs (range).

Domain

The first column in a mapping diagram representing input values in a relation or function.

Range

The second column in a mapping diagram representing the output values in a relation or function.