Predicate Logic and Popular Culture (Part 5): Rickroll

This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Let $G(x,t)$ be the proposition “I am going to do $x$ at time $t$ .” Translate the logical statement

$\forall t \ge 0 \lnot(G(\hbox{give you up},t) \lor G(\hbox{let you down},t) \lor G(\hbox{run around},t) \lor G(\hbox{desert you},t))$ ,

where the domain is all times and time $0$ is now.

By De Morgan’s Laws, this can be rewritten as

$\forall t \ge 0 (\lnot G(\hbox{give you up},t) \land \lnot G(\hbox{let you down},t) \land \lnot G(\hbox{run around},t) \land \lnot G(\hbox{desert you},t))$ ,

which matches the first line in the chorus of the Internet’s most infamous song.

Predicate Logic and Popular Culture (Part 4): A Streetcar Named Desire

Let $D(x,y,t)$ be the proposition “ $x$ depends on $y$ at time $t$ .” Translate the logical statement

$\forall t \le 0 H(\hbox{I},\hbox{kindness of strangers},t)$ ,

where the domain is all times and time $0$ is now.

The clunky way of translating this into English is, “For all times now and in the past, I depended on the kindness of strangers.” This was one of the American Film Institute’s Top 100 lines in the movies, from A Streetcar Named Desire.

Predicate Logic and Popular Culture (Part 3): Casablanca

Let $H(x,y,t)$ be the proposition “ $x$ has $y$ at time $t$ .” Translate the logical statement

$\forall t \ge 0 H(\hbox{We},\hbox{Paris},t)$ ,

where the domain is all times and time $0$ is now.

The clunky way of translating this into English is, “For all times now and in the future, we will have Paris.” Of course, this sounds a whole lot better when Humphrey Bogart says it.

Predicate Logic and Popular Culture (Part 2)

Let $p$ be the proposition “You can write in the proper way,” let $q$ be the proposition “You know how to conjugate,” and let $r$ be the proposition “People mock you online.” Express the implication

$\lnot (p \land q) \Longrightarrow r$

in ordinary English.

By De Morgan’s Laws, the implication could also be written as

$(\lnot p \lor \lnot q) \Longrightarrow r$ ,

thus matching the opening two lines from Weird Al Yankovic’s Word Crimes (a parody of Robin Thicke’s Blurred Lines).

Predicate Logic and Popular Culture (Part 1)

I’ll begin with a few simple examples to illustrate propositional logic.

Let $p$ be the proposition “I am a crook.” Express the negation $\lnot p$ in ordinary English.

Naturally, the negation is one of the most famous utterances in American political history.

Let $p$ be the proposition “She’s cheer captain,” and let $q$ be the proposition “I’m on the bleachers.” Express the conjunction

$p \land q$

in ordinary English.

I could have picked just about anything from popular culture to illustrate this idea, but my choice was Taylor Swift’s biggest hit as a country artist (before she switched to pop). The lyric in question is part of the song’s pre-chorus (for example, at the 39 second mark of the video below).

Let $p$ be the proposition “I will get busy living,” and let $q$ be the proposition “I will get busy dying.” Express the disjunction

$p \lor q$

in ordinary English.

Again, I could have picked almost anything to illustrate disjunctions. My choice comes from a famous scene from The Shawshank Redemption (at the 2:53 mark of the video below — warning, PG language in the rest of the video).

Let $p$ be the proposition “You build it,” and let $q$ be the proposition “He will come.” Express the implication

$p \Longrightarrow q$

in ordinary English.

Of course, this is the famous catchphrase from Field of Dreams.

One more for today:

Let $p$ be the proposition “You want to roam,” and let $q$ be the proposition “You roam.” Express the implication

$p \Longrightarrow q$

in ordinary English.

Though the order of the wording is different, this implication is part of the chorus of one of the biggest hits by the B-52s.

Mathematics and Neuroscience

I stumbled across the following engaging and readable Q&A with Dr. Jonathan Pillow, a professor at Princeton who’s studying how the brain works using mathematics and statistics. I thought that this might be appropriate way of engaging students who think that the study of mathematics is utterly unimportant.

http://www.princeton.edu/main/news/archive/S43/89/07C16/

Careers in Industry for Math Majors

I recently came across an excellent article promoting internships from math majors who would like to use their quantitative skills in an industrial setting (as opposed to an academic setting). The concluding paragraph:

Faculty will continue to train students for academic careers. Some will pursue tenure-track positions in the institutions of their choice, but an increasing number of our students will take positions very different from our own. Let’s learn about those options and share them with our students. Then, when a student takes a good job and enjoys a successful career, let’s call that a win.

Here’s the full article: http://www.americanscientist.org/blog/pub/internships-connect-math-students-to-new-career-paths

My Mathematical Magic Show: Part 3c

Last March, on Pi Day (March 14, 2015), I put together a mathematical magic show for the Pi Day festivities at our local library, compiling various tricks that I teach to our future secondary teachers. I was expecting an audience of junior-high and high school students but ended up with an audience of elementary school students (and their parents). Still, I thought that this might be of general interest, and so I’ll present these tricks as well as the explanations for these tricks in this series. From start to finish, this mathematical magic show took me about 50-55 minutes to complete. None of the tricks in this routine are original to me; I learned each of these tricks from somebody else.

In the last couple of posts, I discussed a trick for predicting the number of triangles that appear when a convex $x-$ gon with $y$ points in the middle is tesselated. Though I probably wouldn’t do the following in a magic show (for the sake of time), this is a natural inquiry-based activity to do with pre-algebra students in a classroom setting (as opposed to an entertainment setting) to develop algebraic thinking. I’d begin by giving the students a sheet of paper like this:

trianglechart

Then I’ll ask them to start on the left box. I’ll tell them to draw a triangle in the box and place one point inside, and then subdivide into smaller triangles. Naturally, they all get 3 triangles.

Then I ask them to repeat if there are two points inside. Everyone will get 5 triangles.

Then I ask them to repeat until they can figure out a pattern. When they figure out the pattern, then they can make a prediction about what the rest of the chart will be.

Then I’ll ask them what the answer would be if there were 100 points inside of the triangle. This usually requires some thought. Eventually, the students will get the pattern $T = 2P+1$ for the number of triangles if the initial figure is a triangle.

Then I’ll repeat for a quadrilateral (with four sides instead of three). After some drawing and guessing, the students can usually guess the pattern $T=2P+2$ .

Then I’ll repeat for a pentagon. After some drawing and guessing, the students can usually guess the pattern $T=2P+3$ .

Then I’ll have them guess the pattern for the hexagon without drawing anything. They’ll usually predict the correct answer, $T = 2P+4$ .

What about if the outside figure has 100 sides? They’ll usually predict the correct answer, $T = 2P+98$ .

What if the outside figure has $N$ sides? By now, they should get the correct answer, $T = 2P + N - 2$ .

This activity fosters algebraic thinking, developing intuition from simple cases to get a pretty complicated general expression. However, this activity is completely tractable since it only involves drawing a bunch of figures on a piece of paper.

My Mathematical Magic Show: Part 3b

This is a magic trick that my math teacher taught me when I was about 13 or 14. I’ve found that it’s a big hit when performed for grade-school children. Here’s the patter:

Magician: Tell me a number between 5 and 10.

Child: (gives a number, call it $x$ )

Magician: On a piece of paper, draw a shape with $x$ corners.

Child: (draws a figure; an example for $x=6$ is shown)

Important Note: For this trick to work, the original shape has to be convex… something shaped like an L or M won’t work. Also, I chose a maximum of 10 mostly for ease of drawing and counting (and, for later, calculating).

Magician: Tell me another number between 5 and 10.

Child: (gives a number, call it $y$ )

Magician: Now draw that many dots inside of your shape.

Child: (starts drawing $y$ dots inside the figure; an example for $y = 7$ ) While the child does this, the Magician calculates $2y + x - 2$ , writes the answer on a piece of paper, and turns the answer face down.

Magician: Now connect the dots with lines until you get all triangles. Just be sure that no two lines cross each other.

Child: (connects the dots until the shape is divided into triangles; an example is shown)

Magician: Now count the number of triangles.

Child: (counts the triangles)

Magician: Was your answer… (and turns the answer over)?

The reason this magic trick works so well is that it’s so counter-intuitive. No matter what convex $x-$ gon is drawn, no matter where the $y$ points are located, and no matter how lines are drawn to create triangles, there will always be $2y + x - 2$ triangles. For the example above, $2y+x-2 = 2\times 7 + 6 - 2 = 18$ , and there are indeed $18$ triangles in the figure.

This trick works by counting the measures of all the angles in two different ways.

Method #1: If there are $T$ triangles created, then the sum of the measures of the angles in each triangle is $180$ degrees. So the sum of the measures of all of the angles must be $180 T$ degrees.

Method #2: The sum of the measures of the angles around each interior point is $360$ degrees. Since there are $y$ interior points, the sum of these angles is $360y$ degrees.

The measures of the remaining angles add up to the sum of the measures of the interior angles of a convex polygon with $x$ sides. So the sum of these measures is $180(x-2)$ degrees.

These two different ways of adding the angles must be the same. In other words, it must be the case that

$180T = 360y + 180(x-2)$ ,

$T = 2y + x - 2$ .

I’m often asked why it was important to choose a number between 5 and 10. The answer is, it’s not important. The trick will work for any numbers as long as there are at least three sides of the polygon. However, in a practical sense, it’s a good idea to make sure that the number of sides and the number of points aren’t too large so that the number of triangles can be counted reasonably quickly.

After explaining how the trick works, I’ll again ask a child to stand up and play the magician, repeating the trick that I just did, before I move on to the next trick.

My Mathematical Magic Show: Part 3a

For my second trick, I’ll show something that my math teacher taught me when I was about 13 or 14. Everyone in the audience has a piece of paper and a pen or pencil. Here’s the patter:

Magician: Tell me a number between 5 and 10.

Child #1: (gives a number, call it $x$ )

Magician: On a piece of paper, draw a shape with $x$ corners. Don’t draw something really, really tiny… make sure it’s big enough to see well.

Audience: (draws a figure; an example for $x=6$ is shown) The Magician also draws this figure on the board.

Important Note: For this trick to work, the original shape has to be convex… something shaped like an L or M won’t work. Also, I chose a maximum of 10 mostly for ease of drawing and counting (and, for later, calculating).

Magician: Tell me another number between 5 and 10.

Child #2: (gives a number, call it $y$ )

Magician: Now draw that many dots inside of your shape.The Magician also draws $y$ dots inside the figure on the board, an example for $y = 7$ is shown.

Audience: (starts drawing $y$ dots inside the figure) The Magician also calculates $2y + x - 2$ and says, “Now while you’re doing that, I’m going to write a secret number on the board,” discreetly writes the answer on the board, and then covers up the answer with a piece of paper and some adhesive tape.

Magician: Now connect the dots with lines until you get all triangles. Just be sure that no two lines cross each other. For example, your figure could look like this:

Audience: (quietly connects the dots until the shape is divided into triangles)

Magician: Now count the number of triangles.

Audience: (counts the triangles)

Magician: Was your answer… (removes the adhesive tape and displays the answer)?

In tomorrow’s post, I’ll explain why this trick works.