| 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 \(x_n\) |
| 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. |
| piecewise rule definition | \(f(x) = \begin{cases} x & \text{if~}x \geq 0 \\ -x & \text{if~}x<0\end{cases}\) | Define \(f\) of \(x\) to be \(x\) when \(x\) is nonnegative and to be \(-x\) when \(x\) is negative |
| 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 \(9\) |
Recall our representation of Netflix users’ ratings of movies as \(n\)-tuples, where \(n\) is the number of movies in the database. Each component of the \(n\)-tuple is \(-1\) (didn’t like the movie), \(0\) (neutral rating or didn’t watch the movie), or \(1\) (liked the movie).
Consider the ratings \(P_1 = (-1, 0, 1)\), \(P_2 = (1, 1, -1)\), \(P_3 = (1, 1, 1)\), \(P_4 = (0,-1,1)\)
Which of \(P_1\), \(P_2\), \(P_3\) has movie preferences most similar to \(P_4\)?
One approach to answer this question: use functions to define distance between user preferences.
For example, consider the function \(d_0: \phantom{the Cartesian product of the set of ratings on 3 movies with itself} \to \phantom{\mathbb{R}}\) given by \[d_0 (~(~ (x_1, x_2, x_3), (y_1, y_2, y_3) ~) ~) = \sqrt{ (x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 -y_3)^2}\]
Extra example: A new movie is released, and \(P_1\) and \(P_2\) watch it before \(P_3\), and give it ratings; \(P_1\) gives and \(P_2\) gives . Should this movie be recommended to \(P_3\)? Why or why not?
Extra example: Define a new function that could be used to compare the \(4\)-tuples of ratings encoding movie preferences now that there are four movies in the database.
| 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 \(x_n\) |
| 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. |
| piecewise rule definition | \(f(x) = \begin{cases} x & \text{if~}x \geq 0 \\ -x & \text{if~}x<0\end{cases}\) | Define \(f\) of \(x\) to be \(x\) when \(x\) is nonnegative and to be \(-x\) when \(x\) is negative |
| 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 \(9\) |
Recall our representation of Netflix users’ ratings of movies as \(n\)-tuples, where \(n\) is the number of movies in the database. Each component of the \(n\)-tuple is \(-1\) (didn’t like the movie), \(0\) (neutral rating or didn’t watch the movie), or \(1\) (liked the movie).
Consider the ratings \(P_1 = (-1, 0, 1)\), \(P_2 = (1, 1, -1)\), \(P_3 = (1, 1, 1)\), \(P_4 = (0,-1,1)\)
Which of \(P_1\), \(P_2\), \(P_3\) has movie preferences most similar to \(P_4\)?
One approach to answer this question: use functions to define distance between user preferences.
For example, consider the function \(d_0: \phantom{the Cartesian product of the set of ratings on 3 movies with itself} \to \phantom{\mathbb{R}}\) given by \[d_0 (~(~ (x_1, x_2, x_3), (y_1, y_2, y_3) ~) ~) = \sqrt{ (x_1 - y_1)^2 + (x_2 - y_2)^2 + (x_3 -y_3)^2}\]
Extra example: A new movie is released, and \(P_1\) and \(P_2\) watch it before \(P_3\), and give it ratings; \(P_1\) gives and \(P_2\) gives . Should this movie be recommended to \(P_3\)? Why or why not?
Extra example: Define a new function that could be used to compare the \(4\)-tuples of ratings encoding movie preferences now that there are four movies in the database.