We use square brackets and parentheses for both written algebaic notation and graphical notation.

We have a very popular alternative for the graphical notation.

• solid or closed dots are often used in place of square brackets to show a number is included in an interval.
• hollow or open dots are often used in place of parentheses to show a number is not included in an interval.

Closed Interval

Half Open Half Close Interval

Half Open Half Close Interval

Open Interval

Union

\[ A = [-4, -2] \cup \{ 2 \} \cup (3, 6] \]

Select all valid statements.

\(A\) is bounded from below \(A\) is bounded from above \(A\) is bounded \(A\) is unbounded from below \(A\) is unbounded from above \(A\) is unbounded

Union

\[ B = (-\infty , -1] \cup (4, 8) \]

Select all valid statements.

\(B\) is bounded from below \(B\) is bounded from above \(B\) is bounded \(B\) is unbounded from below \(B\) is unbounded from above \(B\) is unbounded

Union

\[ C = (-7, -4] \cup \{ -1, 1 \} \cup (3, \infty ) \]

Select all valid statements.

\(C\) is bounded from below \(C\) is bounded from above \(C\) is bounded \(C\) is unbounded from below \(C\) is unbounded from above \(C\) is unbounded

Mathematicians

• Science is the search and identification of structure.
• Mathematics is a language we use to describe and explain our scientific discoveries.

Mathematicians are scientists.

They are searching for structure.

As we accumulate descriptions of structure, we will notice commonalities. We will describe these commonalities and we will organizes our thoughts about this common structure.

We will use mathematics to describe this common structure. We call this abstraction.

This process of description and organization naturally makes our brains think of these descriptions as entities themselves. The language helps us think about structure and becomes structure itself.

Algebra is our first experience with this idea of abstraction.

Mathematicians are looking for structure inside the language we have invented to describe structure.

Mathematics is a language that lends itself to abstract thoughts, which leads to more structure, which leads to more mathematics.

Mathematicians are people who explore this structure.