Predicate Logic and Popular Culture (Part 179): Harry Potter

Let $P$ be the set of all people, let $W(x)$ be the proposition “ $x$ is a witch,” let $Z(x)$ be the proposition “ $x$ is a wizard,” let $B(x)$ be the proposition “ $x$ went bad,” and let $S(x)$ be the proposition “ $x$ was in Slytherin.” Translate the logical statement

$\lnot \exists x \in P( (W(x) \lor Z(x)) \land B(x) \land ~S(x))$ .

This matches matches a line from “Harry Potter and the Sorcerer’s Stone.”

Context: 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.

Predicate Logic and Popular Culture (Part 178): Lauryn Hill

Let $P$ be the set of all people, and let $f(x)$ measure how much $x$ loves you.

$\lnot \exists x \in P(f(x) > f(I))$ .

This matches matches a line from “Ex Factor.” This is also a nice exercise in predicate logic, as the statement is also logically equivalent to

$\forall x \in P(f(x) \le f(I))$ .

Predicate Logic and Popular Culture (Part 177): Shakira

Let $P$ be the set of all people, let $L(x)$ be the proposition “ $x$ learns,” and let $W(x)$ be the proposition “ $x$ gets it wrong.” Translate the logical statement

$\lnot \exists x \in P(L(x) \land \lnot W(x))$ .

This matches this quote from “Try Everything” from “Zootopia.” This is also a nice exercise in predicate logic, as the statement is also logically equivalent to

$\forall x \in P(\lnot L(x) \lor W(x))$

$\forall x \in P(L(x) \Rightarrow W(x))$ .

Predicate Logic and Popular Culture (Part 176): Game of Thrones

Let $p$ be the proposition “You play the game of thrones,” let $q$ be the proposition “You win,” and let $r$ be the proposition “You die.” Translate the logical statement

$p \Rightarrow (q \lor r)$ .

This matches this quote from “Game of Thrones.”

Predicate Logic and Popular Culture (Part 175): Diana Ross and the Supremes

Let $T(t)$ be the proposition “We are together at time $t$ .” Translate the logical statement

$\exists t > 0 (T(t))$ .

This matches this classic by Diana Ross and the Supremes.

Predicate Logic and Popular Culture (Part 174): Rocky IV

Let $p$ be the proposition “He dies.” Translate the logical statement

$p \Rightarrow p$ .

This famous tautology was uttered by Ivan Drago in Rocky IV.

Predicate Logic and Popular Culture (Part 173): Alicia Keys

Let $P$ be the set of all people, let $T$ be the set of all things, and let $W(x,y)$ be the proposition “ $x$ wants $y$ .” Translate the logical statement

$\exists x \in P \forall y \in T (W(x,y))$ .

Of course, this matches the first line in the chorus of this popular Alicia Keys song.

Engaging students: Using a truth table

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 comes from my former student Chris Brown. His topic, from Geometry: using a truth table.

How can this topic be used in your students’ future courses in mathematics or science?

Truth tables apply directly to the field in Computer Science, as in its essence, it runs on Boolean logic. Boolean logic simply means that everything has a result of True or False. This can be seen explicitly when dealing with logic gates, which are different paths that a computer program follows as it tests whether inputs are true or false based on given conditions. Based on the results, the program will continue to run, testing different cases, based on each result in a complex chain of tests. For example, for a simple program, let’s say you may input any integer, n, between 10 and 20 inclusive. If the number is divisible by 2, then it will compute n divided by 2. If the number is not divisible by 2, then it will return the original number. Then, if the resulting number is divisible by 2 as well, it will once again compute n divided by 2. If the resulting number is not divisible by 2, then it will return the resulting number. This sequence of tests follows the conditional statement, “If an integer between 10 and 20 inclusive is divisible by 2, and it’s resulting value is also divisible by 2, then the chosen integer has 2² within its prime factorization.” For the “and” truth table: if the integer chosen was 10, we see the True & False = False case; if the integer 16 was chosen, we see the True & True = True case; if the integer 19 was chosen, we see the False & False = False case. With variations and chains of logic gates, Computer Science has every single type of truth table embedded within the Boolean logic it uses.

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

Logical humor has often been used in the more intelligent based humor of popular culture, and truth tables and arguments are even more so apart of this. In the movie “Get Smart,” released in the year 2008, features a quirky, and humorous data analyst named Maxwell Smart who by an odd turn of events was promoted to field agent. On one of Smart’s missions to infiltrate the enemy base, he, Siegfried, and Shtarker wittingly enters into a logical argument that is a beautifully crafted logical argument. I have written the lines below.

Smart: I understand that you are the man to see if someone is interested in acquiring items of a nuclear nature

Siegfried: How do I know you are not Control

Smart: If I were Control, you would already be dead

Siegfried: If you were Control, you would already be dead

Smart: Since Neither of us are dead, so I guess I am not Control

Shtarker: That actually makes sense!

While this is not an example of a truth table per say, truth tables and propositional logic was the foundation of how this argument was created. What we see in lines 3-5 is the following propositional formula:

((p → q) ∧ (p → s))

Such that:

p = Smart being Control

q = Siegfried Being Dead

s = Smart Being Dead

By viewing the truth table, we see that when q and s are false, then p must be false; as stated in Line 5 of the movie.

How can technology be used to effectively engage students with this topic?

The technology tool that I found was listed on the Stanford University website and is one that the students can easily use to check over their work. The website, attached below, allows students to enter in their propositional logic formulas for any complex length and has functionality for all necessary, binary logical operators. The site also allows for the usage of many logical expressions, not just 2. Inputting the formulas is very user friendly and allows for multiple representations of each logical operator. For instance, “or” can be represented by “\/” and also “or,” and can even both be used within the same formula chain. If a character or statement is used that the system does not recognize, the system will highlight the symbol in red and say, “illegal character,” which I personally find easily understandable for all ages. What I love most about this website is that as the formula is being entered, the student is able to see the table being created as it is being entered.

http://web.stanford.edu/class/cs103/tools/truth-table-tool/

Engaging students: Using Pascal’s triangle

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

This student submission comes from my former student Rachel Delflache. Her topic, from Precalculus: using Pascal’s triangle.

How does this topic expand what your students would have learned in previous courses?

In previous courses students have learned how to expand binomials, however after $(x+y)^3$ the process of expanding the binomial by hand can become tedious. Pascal’s triangle allows for a simpler way to expand binomials. When counting the rows, the top row is row 0, and is equal to one. This correlates to $(x+y)^0 =1$ . Similarly, row 2 is 1 2 1, correlating to $(x+y)^2 = 1x^2 + 2xy + 1y^2$ . The pattern can be used to find any binomial expansion, as long as the correct row is found. The powers in each term also follow a pattern, for example look at $(x+y)^4$ :

$1x^4y^0 + 4x^3y^1 + 6x^2y^2 + 4x^1y^3 + 1x^0y^4$

In this expansion it can be seen that in the first term of the expansion the first monomial is raised to the original power, and in each term the power of the first monomial decreases by one. Conversely, the second monomial is raised to the power of 0 in the first term of the expansion, and increases by a power of 1 for each subsequent term in the expansion until it is equal to the original power of the binomial.

Sierpinski’s Triangle is triangle that was characterized by Wacław Sieriński in 1915. Sierpinski’s triangle is a fractal of an equilateral triangle which is subdivided recursively. A fractal is a design that is geometrically constructed so that it is similar to itself at different angles. In this particular construction, the original shape is an equilateral triangle which is subdivided into four smaller triangles. Then the middle triangle is whited out. Each black triangle is then subdivided again, and the patter continues as illustrated below.

Sierpinski’s triangle can be created using Pascal’s triangle by shading in the odd numbers and leaving the even numbers white. The following video shows this creation in practice.

What are the contributions of various cultures to this topic?

The pattern of Pascal’s triangle can be seen as far back as the 11th century. In the 11th century Pascal’s triangle was studied in both Persia and China by Oman Khayyam and Jia Xian, respectively. While Xian did not study Pascal’s triangle exactly, he did study a triangular representation of coefficients. Xian’s triangle was further studied in 13th century China by Yang Hui, who made it more widely known, which is why Pascal’s triangle is commonly called the Yanghui triangle in China. Pascal’s triangle was later studies in the 17th century by Blaise Pascal, for whom it was named for. While Pascal did not discover the number patter, he did discover many new uses for the pattern which were published in his book Traité du Triangle Arithméthique. It is due to the discovery of these uses that the triangle was named for Pascal.

Reference:
• https://en.wikipedia.org/wiki/Pascal%27s_triangle
• http://mathforum.org/workshops/usi/pascal/images/fill.comb.gif
• https://www.britannica.com/biography/Blaise-Pascal#toc445406main
• https://en.wikipedia.org/wiki/Sierpinski_triangle

Engaging students: Using a truth table

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

This student submission comes from my former student Saundra Francis. Her topic, from Geometry: using a truth table.

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

An activity that can be done with truth tables is notice and wonder. You could give students an example of a truth table, say finding the results of p and q, and have them write down what they notice and what they wonder about the table. This can lead to a great discussion about what they notice about the truth table and how they think it works. Some students might be able to figure out what the table represents or it could help display misconceptions before the lesson starts. Students can then present their wonders, which you can make sure to answers by the end of the lesson. This activity is a good way to discover if students have prior knowledge of the concept and you can see what they hope to have learned by the end of the lesson.

How has this topic appeared in pop culture?

In this Alice in Wonderland clip where Alice attends the Mad Hatter’s tea party we notice the importance of truth tables and logic to distinguish what the Mad Hatter, March Hare, and the Dormouse are discussing. Alice is using logic to discover what the Mad Hatter means and argue with him about his logic. Once you have students watch this video you can ask students to share examples of where logic is displayed. You can choose one example to transfer into a truth table later in the lesson to show students that we can use the logic displayed outside of the classroom. This will engage students and display the importance of using logic to distinguish what others mean.

What interesting things can you say about the people who contributed to the discovery and/or development of this topic?

Bertrand Russell, one of the contributors that made truth tables as we use them today, worked within the philosophical branch that combined mathematics and logic. Russell was born in Trelleck, United Kingdom in 1872. He attended Trinity College in Cambridge achieved a First Class distinction in philosophy. In 1903 Russell wrote The Principles of Mathematics with Dr. Allan Whitehead. This book extended the work of Peano and Frege on mathematical logic. This was the book where he contributed to truth tables. In 1918 Russell was imprisoned for six months due to a pacifist article he wrote. While incarcerated he wrote Introduction to Mathematical Philosophy. Russell received the Nobel Prize in literature in 1950 in recognition of his thoughts on freedom of thought and humanitarian ideals. He is known for his humorous and controversial writing.

References
1. https://www.nobelprize.org/nobel_prizes/literature/laureates/1950/russell-bio.html
2. https://www.teacherspayteachers.com/FreeDownload/Notice-and-Wonder-Truth-Table-2125723
3. https://www.youtube.com/watch?v=1oupIOmnLJs