Definitions set prereqs

Term Notation Example(s) We say in English …
all reals \(\mathbb{R}\) The (set of all) real numbers (numbers on the number line)
all integers \(\mathbb{Z}\) The (set of all) integers (whole numbers including negatives, zero, and positives)
all positive integers \(\mathbb{Z}^+\) The (set of all) strictly positive integers
all natural numbers \(\mathbb{N}\) The (set of all) natural numbers. Note: we use the convention that \(0\) is a natural number.

Definitions functions prereqs

Term Notation Example(s) We say in English …
sequence \(x_1, \ldots, x_n\) A sequence \(x_1\) to \(x_n\)
summation \(\sum_{i=1}^n x_i\) or \(\displaystyle{\sum_{i=1}^n x_i}\) The sum of the terms of the sequence \(x_1\) to \(x_n\)
piecewise rule definition \(f(x) = \begin{cases} \text{rule 1 for } x & \text{when~COND 1} \\ \text{rule 2 for } x & \text{when COND 2}\end{cases}\) Define \(f\) of \(x\) to be the result of applying rule 1 to \(x\) when condition COND 1 is true and the result of applying rule 2 to \(x\) when condition COND 2 is true. This can be generalized to having more than two conditions (or cases).
function application \(f(7)\) \(f\) of \(7\) or \(f\) applied to \(7\) or the image of \(7\) under \(f\)
\(f(z)\) \(f\) of \(z\) or \(f\) applied to \(z\) or the image of \(z\) under \(f\)
\(f(g(z))\) \(f\) of \(g\) of \(z\) or \(f\) applied to the result of \(g\) applied to \(z\)
absolute value \(\lvert -3 \rvert\) The absolute value of \(-3\)
square root \(\sqrt{9}\) The non-negative square root of \(9\)

Pro-tip: the meaning of two vertical lines \(| ~~~ |\) depends on the data-types of what’s between the lines. For example, when placed around a number, the two vertical lines represent absolute value. We’ve seen a single vertial line \(|\) used as part of set builder definitions to represent “such that”. Again, this is (one of the many reasons) why is it very important to declare the data-type of variables before we use them.