Predicate Logic and Popular Culture (Part 35): Elvis

Let $Y(x)$ be the proposition “You are $x$ .” Express the logical statement

$Y(\hbox{a hound dog}) \land \forall x(x \ne \hbox{a hound dog} \Rightarrow \lnot Y(x))$

into ordinary English, where the domain for $x$ are personal attributes.

Perhaps the shortest way to write this would be “You are only a hound dog,” but it’s much catchier when sung by Elvis.

Context: Part of a 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 some time 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.

Predicate Logic and Popular Culture (Part 34): The Beach Boys

Let $W(x,y)$ be the proposition “I wish that $x$ could be $y$ .” Translate the logical statement

$\forall x W(x, \hbox{a California girl})$

into plain English, where the domain for $x$ is all girls.

The simple way to translate this statement is “I wish that all girls could be California girls,” nearly matching the chorus of this classic by the Beach Boys.

Predicate Logic and Popular Culture (Part 33): The Eagles

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

$\exists x \exists y \exists t (0 \le t \le T \land H(x,y,t))$

into plain English, where the domain for $x$ and $y$ are all people, the domain for $t$ is all times, time $0$ is now, and time $T$ is when the night is through.

The simple way to translate this statement is “There are two people so that the first person will hurt the second person at some time between now and when the night is through.” A somewhat briefer way of expressing this thought is made in the first line of this popular song by The Eagles.

Predicate Logic and Popular Culture (Part 32): The Rolling Stones

Let $p$ be the proposition “I can get satisfaction.” Translate the logical statement $\lnot p$ into plain English.

The simple way to translate this statement is “I cannot get satisfaction.” The popular, though grammatically incorrect, way of expression this sentiment was made popular by the Rolling Stones.

Part of a 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.

Engaging students: Combinations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Taylor Vaughn. Her topic, from probability: combinations.

How could you as a teacher create an activity or project that involves your topic?

One interesting way that I thought about introducing combinations is bringing in combinations that students do often, but do not really think about. When it is known that the family is going on vacation, as a girl, the first thought is “what am I going to wear?” Being a girl, I was always told that I cant pack as much as I wanted to because I also wanted to bring extra clothes just in case I didn’t want to wear what I had planned for that day. One activity I thought bout is actually bring in a suitcase to class with clothes and try and plan a 3 day vacation and figure out how we, as a class, was going to pack this suitcase. I could include different scenarios such as, if the hotel has a laundry room, and how would being able to wash clothes and put them back in the suitcase change how we pack. Also, what happens if we add shoes and socks? How would this change affect the number of combinations we can have? I think it would be really cool for students to touch and play and bring in ideas that they don’t necessarily think has anything to do with math.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Everyone listens to music, but there are so many different types of genres, artists, and songs. Have you ever thought, “Will we ever run out of new music?” Well someone by the name of Michael has. He has done the research of what others say about the math of the order of the notes and how many combinations of these notes can we get that will create a new song

One activity that could be done after the video is given 8 notes, how many different measures could students in the class come up with. Then the whole class could see how many people got the same measure or did everyone get something completely different. Then you could also ask “Did we cover all the possibilities? How do we know? How can we show this mathematically?” Lastly, if there are so many possibilities, why are there so many songs with the melodies? There is a video that has one melody and sings a lot of songs to that one melody. (PG-13 Warning: gratuitous cursing near the end of the video.)

The one thing I didn’t like about the first video is the length and he makes connections about songs that are really outdated. SO this video has songs that will relate closer to this generation of students.

How can this topic be used in your students’ future courses in mathematics or science? In school, students that didn’t like math the way I did always asked, “Well when will we ever use this again?” Well even though we use combinations more than we think, it can also be used in later math classes. Ever thought that combinations had anything to do with Pascal’s triangle? What is Pascal’s triangle, you may ask? Well according to Math Is Fun, it is a pattern of numbers where the starting number is 1 and “each number is the two numbers above it added together (except for the edges, which are all “1”).” Well what if you are asked to find entry 20 of the triangle. The one thing I would do would keep the pattern of the triangle and write out all the entries until I got to that number, but using combinations you can get any entry you would like without writing all the entries out. The formula is where n is the row and k is where it is in the nth row.

$\displaystyle {n \choose k} = \frac{n!}{k! (n-k)!}$

While teaching this, I would definitely talk about factorial and how it relates to the lesson.
References

Stevens, Michael. “Will We Ever Run Out of New Music?” YouTube. YouTube, 20 Nov. 2012. Web. 02 Sept. 2015.

The Axis of Awesome. “4 Chords.” YouTube. YouTube, 20 July 2011. Web. 02 Sept. 2015.

Pierce, Rod. “Pascal’s Triangle” Math Is Fun. Ed. Rod Pierce. 30 Mar 2014. 3 Sep 2015 http://www.mathsisfun.com/pascals-triangle.html

Undo It

One of the basic notions of functions that’s taught in Precalculus and in Discrete Mathematics is the notion of an inverse function: if $f: A \to B$ is a one-to-one and onto function, then there is an inverse function $f^{-1}: B \to A$ so that

$f^{-1}(f(a)) = a$ for all $a \in A$ and

$f(f^{-1}(b)) = b$ for all $b \in B$ .

If $A = B = \mathbb{R}$ , this is commonly taught in high school as a function that satisfies the horizontal line test.

In other words, if the function $f$ is applied to $a$ , the result is $f(a)$ . When the inverse function is applied to that, the answer is the original number $a$ . Therefore, I’ll tell my class, “By applying the function $f^{-1}$ , we uh-uh-uh-uh-uh-uh-uh-undo it.”

If I have a few country music fans in the class, this always generates a bit of a laugh.

See also the amazing duet with Carrie Underwood and Steven Tyler at the 2011 ACM awards:

Predicate Logic and Popular Culture (Part 31): The Godfather

Let $p$ be the proposition “I took the gun,” and let $q$ be the proposition “I took the cannoli.” Translate the logical statement

$\lnot p \land q$ .

Obviously, this is an allusion to one of the great lines in The Godfather.

Even though this is a simple example, it actually serves a pedagogical purpose (when I first introduce students to propositional logic) by illustrating two important points.

First, there is an order of precedence with $\lnot$ and $\land$ . Specifically, $\lnot p \land q$ means $(\lnot p) \land q$ (“I did not take the gun, and I took the cannoli”) and not $\lnot (p \land q)$ (“It is false that I took both the gun and the cannoli”).

Second, the actual line from The Godfather is not a proposition because both “Leave the gun” and “Take the cannoli” are commands. By contrast, a proposition must be a declarative sentence that is either true or false. That’s why I had to slightly modify the words to “I took the cannoli” instead of “Take the cannoli.”

Context: 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.

I’m afraid that I found plenty more examples from popular culture to illustrate predicate logic, but a month of posts on this topic is probably enough for now. I’ll return to this topic again at some point in the future.

Predicate Logic and Popular Culture (Part 30): The Platters

Let $R(x)$ be the proposition “ $x$ can make all this the world seem right,” and let $B(x)$ “ $x$ can make the darkness bright.” Translate the logical statement

$R(\hbox{you}) \land B(\hbox{you}) \land \forall x(x \ne \hbox{you} \Longrightarrow \lnot (R(x) \lor B(x)))$ ,

where the domain is all people.

The clunky way of translating this into English is, “You can make all this world seem right, you can make the darkness bright, and everyone else can neither make all this world seem right nor make the darkness bright.” Of course, this is the sentiment expressed by the first two lines of this classic by the Platters.

Predicate Logic and Popular Culture (Part 29): Grease

Let $W(x)$ be the proposition “I want $x$ .” Translate the logical statement

$W(\hbox{you}) \land \forall x(x \ne \hbox{you} \Longrightarrow \lnot W(x))$ ,

where the domain is all people.

The clunky way of translating this into English is, “I want you, and I don’t want anyone who isn’t you.” But it sounds a lot better when John Travolta and Olivia Newton-John sing it.

For professional mathematicians (as opposed to students first learning predicate logic), the more compact way of writing this would be

$W(\hbox{you}) \land \exists! x W(x)$ .

Predicate Logic and Popular Culture (Part 28): High School Musical

Let $L(x)$ be the proposition “ $x$ is a star in heaven” and let $R(x)$ be the proposition “We can reach $x$ ”

$\lnot \exists x(\lnot R(x))$ ,

where the domain for $x$ is the stars in heaven.

The clunky way of translating this into English is, “There is not a star in heaven that we cannot reach,” and this double negative appears in the song Breaking Free from High School Musical.

This example gives students a simple practice problem for using De Morgan’s laws to eliminate the double negative:

$\lnot \exists x(\lnot R(x)) \equiv \forall x(\lnot(\lnot R(x))) \equiv \forall x R(x)$ ,

or “We can reach every star in heaven.”