Set Symbols

$\newenvironment {prompt}{}{} \newcommand {\accordion }[0]{} \newcommand {\ungraded }[0]{} \newcommand {\xmDefaultPreamble }[0]{xmPreamble.tex} \newcommand {\xmDefaultPrintstyle }[0]{xmPrintstyle.sty} \newcommand {\todo }[0]{} \newcommand {\oiint }[0]{{\large \bigcirc }\kern -1.56em\iint } \newcommand {\mooculus }[0]{\textsf {\textbf {MOOC}\textnormal {\textsf {ULUS}}}} \newcommand {\sagemath }[0]{\textsf {SageMath}} \newcommand {\surfaceColor }[0]{violet} \newcommand {\surfaceColorTwo }[0]{redyellow} \newcommand {\sliceColor }[0]{greenyellow} \newcommand {\vector }[1]{\left \langle #1\right \rangle } \newcommand {\sectionOutcomes }[0]{}$

vocabulary

Mathematics is a language. It uses symbols and vocabulary and language and pronunciation.

Sets

We will be investigating functions that are defined on sets. Naturally, we have symbols for sets.

Membership: The symbol $\in $ means “is a member of”.

\[ 7 \in \{ 4, 5, 7, 8 \} \]

$7$ is a member of the set $\{ 4, 5, 7, 8 \}$.

People will also say “$7$ is an element of” $\{ 4, 5, 7, 8 \}$.

People will also say “$7$ is in” $\{ 4, 5, 7, 8 \}$.

Subset: The symbol $\subset $ means “is a proper subset of”.

\[ \{ 4, 5 \} \subset \{ 4, 5, 7, 8 \} \]

$\{ 4, 5 \}$ is a subset of the set $\{ 4, 5, 7, 8 \}$.

Every member of $\{ 4, 5 \}$ is also a member of $\{ 4, 5, 7, 8 \}$.

Proper means that the subset is not equal to the larger set.

The symbol $\subseteq $ means “is a subset of”. This allows the possibility of the subset being equal to the larger set.

\[ \{ 4, 5, 7, 8 \} \subseteq \{ 4, 5, 7, 8 \} \]

Union: The symbol $\cup $ stands for “union”.

The union of two sets is another set. The union contains all of the members of the two original sets.

\[ \{ 1, 2, 3 \} \cup \{ 4, 5, 7, 8 \} = \{ 1, 2, 3, 4, 5, 7, 8 \} \]

Intersection: The symbol $\cap $ stands for “intersection”.

The intersection of two sets is another set. The intersection contains all of the members shared by the two original sets.

\[ \{ 1, 2, 3, 4, 5 \} \cap \{ 4, 5, 7, 8 \} = \{ 4, 5 \} \]

Empty Set: The symbol $\emptyset $ stands for “the empty set”.

The empty set is a set. It just contains no members.

Numbers

We have some standard sets of numbers and they have special symbols.

$\mathbb {N}$ : the natural numbers
$\mathbb {Z}$ : the integers
$\mathbb {Q}$ : the rational numbers
$\mathbb {R}$ : the real numbers
$\mathbb {C}$ : the complex numbers

Small

As we move toward Calculus, our attention will focus on “close”... a lot!

The word instantaneous will describe our “close” measurements.

So, we will use our symbols, notation, and language to help us talk about “close”.

We have two symbols from the Greek language that we traditionally use to mean “a small positive number”.

Small

epsilon: $\epsilon $

delta: $\delta $

$\epsilon $ and $\delta $ are used to mean “a very small positive number”.

$\epsilon $ and $\delta $ are how we talk algebraically about “close”.

$\epsilon $ and $\delta $ are not specific numbers with specific numeric values, like $\pi $.

$\epsilon $ and $\delta $ are used to represent very small positive numbers, but not specific small positive numbers.

So, $\epsilon $ and $\delta $ are used as constants - small positive constants.

When we use $\epsilon $ and $\delta $, we are talking about numbers smaller than $0.000000000001$.

$\blacktriangleright $ “small” means “close” to $0$.

A number can be small and negative or a number can be small and positive. Small just means really close to $0$.

$\blacktriangleright $ “big” or “large” means “far way from” to $0$.

A number can be big and negative or a number can be big and positive. Big just means far awy from $0$.

“Big” and “small” are size words. They describe how close a number is to $0$.

Separate from this are “greater than” and “less than”. These are position words.

$\blacktriangleright $ “lesser” or “less than” mean to the left of on the number line. These words describe realtive position.

$\blacktriangleright $ “Greater” or “less than” mean to the right of on the number line. These words describe realtive position.

The number $-7$ is less than the number $3$, but $3$ is smaller than $-7$.

The number $-9$ is less than the number $2$, but $-9$ is bigger than $2$.

ooooo-=-=-=-ooOoo-=-=-=-ooooo
more examples can be found by following this link
More Examples of Real-Valued Functions

2025-07-27 19:15:21