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?.”

Existence Proofs

Source: https://xkcd.com/1856/

How To Use Facebook Emoji to Respond to a Mathematical Proof

Source: https://www.facebook.com/MathWithBadDrawings/photos/a.822582787758549/2865881973428610/?type=3&theater

Adding by a Form of 0 (Part 3)

As part of my discrete mathematics class, I introduce my freshmen/sophomore students to various proof techniques, including proofs about sets. Here is one of the examples that I use that involves adding and subtracting a number twice in the same proof.

Theorem. Let A be the set of even integers, and define

B = \{ n: n = m+1 for some odd integer m\}

Then A = B.

Proof (with annotations). Before starting the proof, I should say that I expect my students to use the formal definitions of even and odd:

  • An integer n is even if n = 2k for some integer k.
  • An integer n is odd if n = 2k+1 for some integer k.

To prove that A = B, we must show that A \subseteq B and B \subseteq A. The first of these tends to trickiest for students.

Part 1. Let n \in A. By definition of even, that means that there is an integer k so that n = 2k.

To show that n \in B, we must show that n = m + 1 for some odd integer m. To this end, notice that n = (n-1) + 1. Thus, we must show that n - 1 is an odd integer, or that n -1 can be written in the form 2k+1. To do this, we add and subtract 1 a second time:

n = 2k

= (2k - 1) + 1

= ([2k - 1 - 1] + 1) + 1

= ([2k-2] + 1) + 1

= (2[k-1] + 1) + 1.

By the closure axioms, k-1 is an integer. Therefore, 2[k-1] + 1 is an odd number by definition of odd, and hence $n \in B$.

The above part of the proof can be a bit much to swallow for students first learning about proofs. For completeness, let me also include Part 2 (which, in my experience, most students can produce without difficulty).

Part 2. Let n \in B, so that n = m + 1 for some odd integer m. By definition of odd, there is an integer k so that $m = 2k+1$. Therefore, n = (2k+1) + 1 = 2k+2 = 2(k+1). By the closure axioms, k +1 is an integer. Therefore, n is even by definition of even, and so we conclude that n \in A.

\square

For what it’s worth, this is the review problems for which I recorded myself talking through the solution for the benefit of my students.

In my opinion, the biggest conceptual barriers in this proof are these steps from Part 1:

2k = (2k - 1) + 1 = ([2k - 1 - 1] + 1) + 1.

These steps are undeniably awkward. Back in high school algebra, students would get points taken off for making the expression more complicated instead of simplifying the answer. But this is the kind of jump that I need to train my students to do so that they can master this technique and be successful in their future math classes.

Predicate Logic and Popular Culture (Part 206): Jack Johnson

Let H be the set of all things, let T be the set of all times, let G(x) be the proposition “x is good,” and let R(x,t) be the proposition “x remains at time t.” Translate the logical statement

\forall x \in H(G(x) Longrightarrow \forall t \in T(R(x,t))).

This matches a line from “Mudfootball” by Jack Johnson.

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 205): Bob Marley

Let T be the set of all things, let L(x) be the proposition “x is a little thing,” and let A(x) be the proposition “x is going to be all right.” Translate the logical statement

\forall x \in T(L(x) \Longrightarrow A(x)).

This matches a line from “Three Little Birds” by Bob Marley.

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 204): Billy Joel

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

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

This matches a line from “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 203): Bill Withers

Let P be the set of all people, let T be the set of all times, let P(x,t) be the proposition “x has pain at time t,” and let S(x,t) be the proposition “x has sorrow at time t.” Translate the logical statement

\forall x \in P( \exists t_1 \in T(P(x,t)) \land \exists t_2 \in T(S(x,t)).

This matches a line from “Lean on Me.” Note: while I think the translation above matches the intent of the song, a case could be made that, literally rendered, the “there exists” symbols should come first — that there’s a single time that everyone has pain at that one time.

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.