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”.

$$7 \in \{ 4, 5, 7, 8 \}$$

People will also say “is an element of”.

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

$$\{ 4, 5 \} \subset \{ 4, 5, 7, 8 \}$$

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 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 measurements.

So, we will use our symbols, notation, 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”.

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

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