News flash: ChatGPT sometimes asserts false statements are true (Part 2)

I feel like I’ve done my good deed for the day by uncovering another instance when ChatGPT claimed a “fact” from the secondary mathematics curriculum that is simply incorrect. This one’s a doozy: it claimed that the binomial coefficent \displaystyle {23 \choose 6} = 100,947 is equal to \displaystyle {22 \choose 5} = 26,334. (The first identity in the first line below is correct; the second one is not.)

Lesson: ChatGPT is a nice tool but you get what you pay for.

Predicate Logic and Popular Culture (Part 297): Megamind

Let p be the statement “This is an Easter Bunny,” let q be the statement “There is a tooth fairy,” and let r be the statement “There is a Queen of England.” Translate the logical statement

\sim p \land \sim q \land \sim r.

This matches a line from the movie “Megamind.”

green line

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.

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 296): Kanye West

Let H be the set of all things, K(x) be the proposition “x kills you,” and let S(x) be the proposition “x makes you stronger.” Translate the logical statement

\forall x \in H( \sim K(x) Rightarrow S(x) ).

This matches the first line of “Stronger” by Kanye West.

green line

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.

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 295): Radiohead

Let p be the statement “I am king” and let q be the statement “You will be first against the wall.” Translate the logical statement

p \Longrightarrow q

This matches a line by from “Paranoid Android” by Radiohead.

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 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 294): Aerosmith

Let P be the set of all people, and let D(x) be the statement “x has dues in life to pay.” Translate the logical statement

\forall x \in P (D(x))

This matches a line by from “Dream On” by Aerosmith.

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 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 293): The Dark Knight

Let p be the statement “You die a hero” and let q be the statement “You live long enough to see yourself become the villian.” Translate the logical statement

(p \land \sim q) \lor (\sim p \land q)

This matches a line by Harvey Dent/Two-Face in “The Dark Knight.”

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 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.

The Incomplete Gamma and Confluent Hypergeometric Functions: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The links below show my series on the incomplete gamma function and the confluent hypergeometric function.

Part 1: Introduction

Part 2: Direct confirmation

Part 3: Start of the derivation

Part 4: Link to Pascal’s triangle

Part 5: Visualization of key property of Pascal’s triangle

Part 6: Formal proof of key property

Part 7: Putting it all together

High School Students Finding New Proofs of Old Theorems (Part 3): Cayley’s Formula

This is a new favorite story to share with students: a high school student recently published an elementary proof of Cayley’s Formula from graph theory.

Professional article in the American Mathematical Monthly (requires a subscription): https://www.tandfonline.com/doi/full/10.1080/00029890.2025.2590423

By coincidence, this is one of the math theorems that was cited in the movie Good Will Hunting.

Predicate Logic and Popular Culture (Part 292): Radiohead

Let H be the set of all things, F(x) be the statement “You feel x,” and T(x) be the statement “x is there. Translate the logical statement

\sim \forall x \in H (F(x) \Longrightarrow T(x))

This matches a line from “There, There” by Radiohead.

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 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 291): Lemonade Mouth

Let H be the set of all things, D(x) be the statement “x is holding back our dreams,” and L(x) be the statement “We are going to let go of x.” Translate the logical statement

\forall x \in H (D(x) \Longrightarrow L(x))

This matches a line from “Somebody” by Lemonade Mouth.

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 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.