In the table below, each row represents a user’s ratings of movies: (check) indicates the person liked the movie, (x) that they didn’t, and \(\bullet\) (dot) that they didn’t rate it one way or another (neutral rating or didn’t watch). Can encode these ratings numerically with \(1\) for (check), \(-1\) for (x), and \(0\) for \(\bullet\) (dot).
| Person | Fyre | Frozen II | Picard | Ratings written as a \(3\)-tuple |
|---|---|---|---|---|
| \(P_1\) | \(\bullet\) | |||
| \(P_2\) | ||||
| \(P_3\) | ||||
| \(P_4\) | \(\bullet\) |
| 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 \(9\) |
| Term | Examples: |
| (add additional examples from class) | |
| set unordered collection of elements | \(7 \in \{43, 7, 9 \}\) \(2 \notin \{43, 7, 9 \}\) |
| repetition doesn’t matter | |
| Equal sets agree on membership of all elements | |
| \(n\)-tuple ordered sequence of elements with \(n\) “slots" (\(n >0\)) | |
| repetition matters, fixed length | |
| Equal \(n\)-tuples have corresponding components equal | |
| string ordered finite sequence of elements each from specified set | |
| repetition matters, arbitrary finite length | |
| Equal strings have same length and corresponding characters equal |
Special cases:
When \(n=2\), the 2-tuple is called an ordered pair.
A string of length \(0\) is called the empty string and is denoted \(\lambda\).
A set with no elements is called the empty set and is denoted \(\{\}\) or \(\emptyset\).
To define sets:
To define a set using roster method, explicitly list its elements. That is, start with \(\{\) then list elements of the set separated by commas and close with \(\}\).
To define a set using set builder definition, either form “The set of all \(x\) from the universe \(U\) such that \(x\) is ..." by writing \[\{x \in U \mid ...x... \}\] or form “the collection of all outputs of some operation when the input ranges over the universe \(U\)" by writing \[\{ ...x... \mid x\in U \}\]
We use the symbol \(\in\) as “is an
element of” to indicate membership in a set.
Example sets: For each of the following,
identify whether it’s defined using the roster method or set builder
notation and give an example element.
\(\{ -1, 1\}\)
\(\{0, 0 \}\)
\(\{-1, 0, 1 \}\)
\(\{(x,x,x) \mid x \in \{-1,0,1\}
\}\)
\(\{ \}\)
\(\{ x \in \mathbb{Z} \mid x \geq 0
\}\)
\(\{ x \in \mathbb{Z} \mid x > 0
\}\)
\(\{\texttt{A},\texttt{C},\texttt{U},\texttt{G}\}\)
\(\{\texttt{A}\texttt{U}\texttt{G},
\texttt{U}\texttt{A}\texttt{G}, \texttt{U}\texttt{G}\texttt{A},
\texttt{U}\texttt{A}\texttt{A}\}\)
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.
In the table below, each row represents a user’s ratings of movies: (check) indicates the person liked the movie, (x) that they didn’t, and \(\bullet\) (dot) that they didn’t rate it one way or another (neutral rating or didn’t watch). Can encode these ratings numerically with \(1\) for (check), \(-1\) for (x), and \(0\) for \(\bullet\) (dot).
| Person | Fyre | Frozen II | Picard | Ratings written as a \(3\)-tuple |
|---|---|---|---|---|
| \(P_1\) | \(\bullet\) | |||
| \(P_2\) | ||||
| \(P_3\) | ||||
| \(P_4\) | \(\bullet\) |
| 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 \(9\) |
| Term | Examples: |
| (add additional examples from class) | |
| set unordered collection of elements | \(7 \in \{43, 7, 9 \}\) \(2 \notin \{43, 7, 9 \}\) |
| repetition doesn’t matter | |
| Equal sets agree on membership of all elements | |
| \(n\)-tuple ordered sequence of elements with \(n\) “slots" (\(n >0\)) | |
| repetition matters, fixed length | |
| Equal \(n\)-tuples have corresponding components equal | |
| string ordered finite sequence of elements each from specified set | |
| repetition matters, arbitrary finite length | |
| Equal strings have same length and corresponding characters equal |
Special cases:
When \(n=2\), the 2-tuple is called an ordered pair.
A string of length \(0\) is called the empty string and is denoted \(\lambda\).
A set with no elements is called the empty set and is denoted \(\{\}\) or \(\emptyset\).
To define sets:
To define a set using roster method, explicitly list its elements. That is, start with \(\{\) then list elements of the set separated by commas and close with \(\}\).
To define a set using set builder definition, either form “The set of all \(x\) from the universe \(U\) such that \(x\) is ..." by writing \[\{x \in U \mid ...x... \}\] or form “the collection of all outputs of some operation when the input ranges over the universe \(U\)" by writing \[\{ ...x... \mid x\in U \}\]
We use the symbol \(\in\) as “is an
element of” to indicate membership in a set.
Example sets: For each of the following,
identify whether it’s defined using the roster method or set builder
notation and give an example element.
\(\{ -1, 1\}\)
\(\{0, 0 \}\)
\(\{-1, 0, 1 \}\)
\(\{(x,x,x) \mid x \in \{-1,0,1\}
\}\)
\(\{ \}\)
\(\{ x \in \mathbb{Z} \mid x \geq 0
\}\)
\(\{ x \in \mathbb{Z} \mid x > 0
\}\)
\(\{\texttt{A},\texttt{C},\texttt{U},\texttt{G}\}\)
\(\{\texttt{A}\texttt{U}\texttt{G},
\texttt{U}\texttt{A}\texttt{G}, \texttt{U}\texttt{G}\texttt{A},
\texttt{U}\texttt{A}\texttt{A}\}\)
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.