AI Tutor
If a recipe tells you to bake at "between 350°F and 375°F," does that include 350°F exactly? In mathematics, sloppy boundaries lead to broken functions and undefined operations.
To prevent ambiguity, mathematicians created three interchangeable dialects to describe sets of numbers: inequalities, intervals, and set-builder notation.

Three Dialects, One Meaning

The domain is the complete set of valid input values (xx), while the range is the complete set of possible output values (yy).
Every continuous group of numbers can be written in inequality notation, interval notation, or set-builder notation without changing its value.
How do you know when to use a square bracket versus a round parenthesis in interval form?

Intervals: Brackets vs. Parentheses

A square bracket ([[ or ]]) indicates that an endpoint is included in the set, corresponding to ≤\le or ≥\ge.
A round parenthesis ((( or ))) indicates that an endpoint is excluded, corresponding to << or >>.
Because infinity (∞\infty) is a concept of unbounded growth rather than a reachable number, infinity always takes a round parenthesis: (−∞,∞)(-\infty, \infty).
What if you want to specify not just the interval boundaries, but also the universe of numbers you are choosing from?

Set-Builder: The Formal Recipe