Introduction to Set Theory and Number Sets

These flashcards cover the fundamental definitions, symbols, and special number sets within set theory as presented in the lecture notes.

Set

A collection of objects or things, usually denoted by a capital letter.

{ }

Curly brackets read as "the set of".

Elements

The objects or members which make up a set.

$$\in$$

Used to mean "is an element of" or "is a member of".

$$\notin$$

Used to mean "is not an element of" or "is not a member of".

$$n(A)$$

The number of elements in the finite set $A$.

Subset ($A \subseteq B$)

$A$ is a subset of $B$ if all elements of $A$ are also elements of $B$.

Equal Sets

Two sets $A$ and $B$ are equal if their elements are exactly the same.

Proper Subset ($A \subset B$)

Every element of $A$ is also an element of $B$, but $A \neq B$.

Universal Set ($U$)

A set that contains all of the elements under consideration.

Empty Set

A set that has no elements, denoted by $\emptyset$ or ${ }$; it is a proper subset of any other set.

Complement of a Set ($A'$)

The set of all elements of $U$ which are not elements of $A$.

$$N$$

The set of all natural or counting numbers $\{0, 1, 2, 3, 4, 5, 6 ......\}$, where $n(N)$ is infinite.

$$Z$$

The set of all integers $\{0, \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6 ......\}$.

$$Z^+$$

The set of all positive integers $\{1, 2, 3, 4, 5, 6 ......\}$.

$$Q$$

The set of all rational numbers which have the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

$$R$$

The set of all real numbers, which are numbers which can be placed on a number line.

$$Q^+$$

The set of positive rational numbers, defined as $\{x | x > 0, x \in Q\}$.

$$R^+$$

The set of positive real numbers, defined as $\{x | x > 0, x \in R\}$.

Interval Notation

A method to quickly describe sets of numbers using mathematical symbols, where $\{x | ......\}$ denotes the set of all $x$ such that a condition is met.