Set Symbols

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vocabulary

Mathematics is a language. It uses symbols, notation, vocabulary, and pronunciation. Just like every language.

Sets

We will be investigating functions that are defined on sets of real numbers. Naturally, mathematics has symbols, notation, vocabulary, and pronunciation 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 $\subseteq $ means “is a subset of”. A subset is a collection of members of another set.

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

Every set is a subset of itself.

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

Subset: The symbol $\subset $ means “is a proper subset of”. A proper subset is a collection of members of a larger set, but not every memeber from the larger set.

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

$\{ 4, 5 \}$ is a proper 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 smaller subset is not equal to the larger set. Something is not included.

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 the idea of “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 often used to represent “a very small positive number”.

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

Not Specific Numbers

$\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 $0.000000000001$.

Big and Small

$\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 away 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.

Left and Right

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

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

Smaller and Less Than

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

Bigger and Less Than

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

Numbers get bigger as they move away from $0$ in either direction.

We will have many situations where numbers approach $-\infty $. That is they get bigger negatively. They don’t get smaller. They head to the left on the number line. They get lesser, just negatively. They get lesser and bigger.

Small and less are synonyms if you are only talking about positive numbers. With negative numbers, the story is much more interesting.

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

2026-03-18 14:23:18