Category: Discrete mathematics

Predicate Logic and Popular Culture (Part 213): Harry Styles

Let L(t) be the statement “We learn at time t,” let B(t) be the statement “We’ve been here at time t,” let T be the set of all times, and let time 0 be now. Translate the logical statement

\forall t \in T(\lnot L(t)) \land \exists t < 0 (B(t)).

This matches a repeated line in “Sign of the Times” by Harry Styles.

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 212): Billy Joel

Let W(t) be the statement “She is a woman to me at time t,” and let T be the set of all times. Translate the logical statement

\forall t \in T(W(t)).

This matches the chorus of “She’s Always a Woman” by Billy Joel.

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 211): Knuckle Puck

Let L(x) be the statement “x lies to me,” and let P be the set of all people. Translate the logical statement

\forall x \in P(L(x)).

This matches a line from the song (and the title of the song) “Everyone Lies to Me” by Knuckle Puck.

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 210): Alan Walker, Sabrina Carpenter & Farruko

Let S(x) be the statement “x can keep me safe,” and let P be the set of all people. Translate the logical statement

S(I) \land \forall x \in P(x \ne I \Rightarrow \lnot S(x)).

This matches a line from the song “On My Way.”

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 209): The Office

Let p be the statement “I am superstitious,” and let q be the statement “I am a little stitious.” Translate the logical statement

\lnot p \land q.

This matches a quote from the popular TV show “The Office.”

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 208): Masashi Kishimoto

Let R(t) be the statement “I run away at time t,” let G(t) be the statement “I go make on my word at time t,” and let T be the set of all times. Translate the logical statement

\forall t \in T (\lnot R(t) \land \lnot G(t)).

This matches a quote from the main character Naruto in Masashi Kishimoto’s anime (also named Naruto). 

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 207): Patrick Rothfuss

Let p be the statement “You tell a story the right way,” and let q be the statement “You are a bit of a liar.” Translate the logical statement

p \Rightarrow q.

This matches a line from the novel “Name of the Wind” by Patrick Rothfuss: “You have to be a bit of a liar to tell a story the right way.” https://www.goodreads.com/quotes/155428-you-have-to-be-a-bit-of-a-liar-to

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.

Is 8,675,309 prime?

This semester, to remind today’s college students of the greatness of the 1980s: I made my class answer the following question on an exam:

Jenny wants to find out if 8,675,309 is prime. In a few sentences, describe an efficient procedure she could use to answer this question.

Amazingly, it turns out that 8,675,309 is a prime number, though I seriously doubt that Tommy Tutone had this fact in mind when he wrote the classic 80s song. To my great disappointment, nobody noticed (or at least admitted to noticing) the cultural significance of this number on the exam.

Naturally, I didn’t expect my students to actually determine this on a timed exam, and I put the following elaboration on the exam:

Although Jenny has a calculator, answering this question would take more than 80 minutes. So don’t try to find out if it’s prime or not! Instead, describe a procedure for answering the question and provide enough details so that Jenny could follow your directions. Since Jenny will need a lot of time, your procedure should be efficient, or as quick as possible (even if it takes hours).

Your answer should include directions for making a certain large list of prime numbers. Be sure to describe the boundaries of this list and how this list can be made efficiently. Hint: We described an algorithm for making such lists of prime numbers in class. (Again, do not actually construct this list.)

I thought it was reasonable to expect them to describe a process for making this determination on a timed exam.  Cultural allusions aside, I thought this was a good way of checking that they conceptually understood certain facts about prime numbers that we had discussed in class:

  • First, to check if 8,675,309 is prime, it suffices to check if any of positive prime numbers less than or equal to \sqrt{8,675,309} \approx 2,945.387\dots are factors of 8,675,309.
  • To make this list of prime numbers, the sieve of Eratosthenes can be employed. Notice that \sqrt{2,945} \approx 54.271\dots, and the largest prime number less than this number is 53. Therefore, to make this list of prime numbers, one could write down the numbers between 2 and 2,945 and then eliminate the nontrivial multiples of the prime numbers 2, 3, 5, 7, 11, \dots 53.
  • If none of the resulting prime numbers are factors of 8,675,309, then we can conclude that 8,675,309 is prime.

I was happy that most of my class got this answer either entirely correct or mostly correct… and I was also glad that nobody suggested the efficient one-sentence procedure “Google Is 8,675,309 prime?.”