Tag: logic

Predicate Logic and Popular Culture (Part 66): West Side Story

Let L(x,t) be the proposition “We find a new way of living at place x and at time t,” and let F(x,t) be the proposition “We find a way of forgiving at place x and t.” Translate the logical statement

\exists x \exists t (L(x,t) \land F(x,t)).

This matches almost perfectly one of the lines from the classic song “Somewhere” from the musical West Side Story.

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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 65): John Philip Sousa

Let S(t) be the proposition “The Stars and Stripes wave at time t.” Translate the logical statement

\forall t (S(t)).

I tried to think of a fitting example for the Fourth of July, but the best that I could find was the closing line of the chorus of the Stars and Stripes Forever.

Which naturally leads me to this amazing version from the 1970s:

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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 64): Abraham Lincoln

Let F(x,t) be the proposition “You can fool x at time t.” Translate the logical statement

\exists t_1 \forall x (F(x,t_1)) \land \exists x_1 \forall t(F(x_1,t)) \land \lnot(\forall x \forall t(F(x,t))).

Of course, this is the famous quote commonly attributed to Abraham Lincoln: “You can fool all of the people some of the time, and some of the people all of the time, but you cannot fool all of the people all of the time.”

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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 63): P. T. Barnum

Let S(x) be the proposition "latex x$ is a sucker,” and let B(x,t) be the proposition “x is born at time t.” Translate the logical statement

\forall t \exists x (S(x) \land B(x,t)).

Naturally, this is the famous quote often attributed to P. T. Barnum: “There’s a sucker born every minute.”

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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 62): George Strait

Let X(x) be the proposition “x is my ex,” and let T(x) be the proposition “x lives in Texas.” Translate the logical statement

\forall x (X(x) \Rightarrow T(x)),

where the domain is all people.

Naturally, this one of the great hits in the storied career of George Strait.

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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.

Engaging students: Recognizing equivalent statements

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 Tiffany Jones. Her topic, from Geometry: recognizing equivalent statements.

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

How has this topic appeared in pop culture?

The topic of recognizing equivalent statements appears in pop culture in Lewis Carroll’s “Alice’s Adventures in Wonderland”.  English mathematician Reverend Charles Lutwidge Dodgson published the work in 1865 under the pseudonym Lewis Carroll.

Reverend Charles Lutwidge Dodgson was a mathematician, so it is not surprising that mathematical concepts appear in his writing. During the tea party scene, examples of logic statements are present. More specifically, how statements and their converse do not have the same meaning or truth-value. The March Hare asks Alice to say what she means, to which she responds with that she means what she says, thinking that to be the same thing. The Hatter disagrees and the conversation continues with three examples to show Alice that the statements are not the same. Here is the text:

Here is the excerpt from the text from chapter seven:

‘Do you mean that you think you can find out the answer to it?’ said the March Hare.

‘Exactly so,’ said Alice.

‘Then you should say what you mean,’ the March Hare went on.

‘I do,’ Alice hastily replied; ‘at least—at least I mean what I say—that’s the same thing, you know.

”Not the same thing a bit!’ said the Hatter. ‘You might just as well say that “I see what I eat” is the same thing as “I eat what I see”!

”You might just as well say,’ added the March Hare, ‘that “I like what I get” is the same thing as “I get what I like”!

”You might just as well say,’ added the Dormouse, who seemed to be talking in his sleep, ‘that “I breathe when I sleep” is the same thing as “I sleep when I breathe”!

”It IS the same thing with you,’ said the Hatter, and here the conversation dropped, and the party sat silent for a minute, while Alice thought over all she could remember about ravens and writing–desks, which wasn’t much.

The lesson on logical statements and truth-values would start with a reading of this section of text or viewing of a clip that keeps the original text.  Take a simple statement and write its converse, inverse, and contrapositive. For example, “I like what I get” becomes “I get what I like”, “I do not like what I do not get”, and “I do not get what I do not like”, respectively.  Discuss the truth-values of each of the statements show that the original and contrapositive are equivalent and that the converse and inverse are equivalent, to help the students see patterns when rewriting a statement.

Then the students will complete a worksheet by MathBits.com to ensure that they understand the process with simple English sentences and to introduce them to the idea with simple mathematical statements. The worksheet includes a portion of the text above for the students’ reference. The worksheet has the students take two simple sentence and write them in the form of “if…, then..”, then the students are to  write their  converse, inverse, and contrapositive. Next, the students compare the truth-values of each statement. Finally, the students are given two mathematical statements and are asked to determine the truth-values.

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

Recognizing equivalent statements appears in analysis courses and courses which proofs are often used. Being able to recognize equivalent statements adds another tool to the the tool box of proof writing.  With Proof writing, sometimes the contrapositive form of the statement is easier to prove than the original statement itself.

For example, in Math 4050 Advanced Study of Secondary Mathematics Curriculum, the proof of the following theorem is easier to prove by contrapositive than just straight on.

If a prime p divides m*n with m and n composite, then p divides m or        p divides n.

The contrapositive, if p does not divide m and p does not divide n with p, m, and n the same as before, then p does not divide m*n, follows easily (with a little clairvoyance) from another theorem for the class.

 

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Carroll, Lewis. Alice’s Adventures in Wonderland. Lit2Go Edition. 1865. Web. <http://etc.usf.edu/lit2go/1/alices-adventures-in-wonderland/&gt;. October 1, 2015.

Carroll, Lewis. “Chapter VII: A Mad Tea-Party.” Alice’s Adventures in Wonderland. Lit2Go Edition. 1865. Web. <http://etc.usf.edu/lit2go/1/alices-adventures-in-wonderland/17/chapter-vii-a-mad-tea-party/&gt;. October 1, 2015.

Roberts, Frederick, and MathBits.com. Alice in Wonderland Worksheet. S.l.: Commission of the European Communities, 1993. Mathbits. Commission of the European Communities. Web. 1 Oct. 2015

University of North Texas course math 4950 Advanced Study of Secondary Mathematics Curriculum lecture Fall 2015 taught Dr. John Quintanilla

 

 

The Shortest Known Paper Published in a Serious Math Journal

Source: http://www.openculture.com/2015/04/shortest-known-paper-in-a-serious-math-journal.html

Euler’s conjecture, a theory proposed by Leonhard Euler in 1769, hung in there for 200 years. Then L.J. Lander and T.R. Parkin came along in 1966, and debunked the conjecture in two swift sentences. Their article — which is now open access and can be downloaded here — appeared in the Bulletin of the American Mathematical Society.

 

Predicate Logic and Popular Culture: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on using examples from popular culture to illustrate principles of predicate logic.

Unlike other series that I’ve made, this series didn’t have a natural chronological order. So I’ll list these by concept illustrated from popular logic.

green lineLogical and \land: Part 1

  • Part 1: “You Belong To Me,” by Taylor Swift
  • Part 21: “Do You Hear What I Hear,” covered by Whitney Houston
  • Part 31: The Godfather (1972)
  • Part 45: The Blues Brothers (1980)
  • Part 53: “What Does The Fox Say,” by Ylvis
  • Part 54: “Billie Jean,” by Michael Jackson

Logical or \lor:

  • Part 1: Shawshank Redemption (1994)

Logical negation \lnot:

  • Part 1: Richard Nixon
  • Part 32: “Satisfaction!”, by the Rolling Stones
  • Part 39: “We Are Never Ever Getting Back Together,” by Taylor Swift

Logical implication \Rightarrow:

  • Part 1: Field of Dreams (1989), and also “Roam,” by the B-52s
  • Part 2: “Word Crimes,” by Weird Al Yankovic
  • Part 7: “I’ll Be There For You,” by The Rembrandts (Theme Song from Friends)
  • Part 43: “Kiss,” by Prince
  • Part 50: “I’m Still A Guy,” by Brad Paisley

For all \forall:

  • Part 3: Casablanca (1942)
  • Part 4: A Streetcar Named Desire (1951)
  • Part 34: “California Girls,” by The Beach Boys
  • Part 37: Fellowship of the Ring, by J. R. R. Tolkien
  • Part 49: “Buy Me A Boat,” by Chris Janson
  • Part 57: “Let It Go,” by Idina Menzel and from Frozen (2013)

For all and implication:

  • Part 8 and Part 9: “What Makes You Beautiful,” by One Direction
  • Part 13: “Safety Dance,” by Men Without Hats
  • Part 16: The Fellowship of the Ring, by J. R. R. Tolkien
  • Part 24 : “The Chipmunk Song,” by The Chipmunks
  • Part 55: The Quiet Man (1952)

There exists \exists:

  • Part 10: “Unanswered Prayers,” by Garth Brooks
  • Part 15: “Stand by Your Man,” by Tammy Wynette (also from The Blues Brothers)
  • Part 36: Hamlet, by William Shakespeare
  • Part 57: “Let It Go,” by Idina Menzel and from Frozen (2013)

Existence and uniqueness:

  • Part 14: “Girls Just Want To Have Fun,” by Cyndi Lauper
  • Part 20: “All I Want for Christmas Is You,” by Mariah Carey
  • Part 23: “All I Want for Christmas Is My Two Front Teeth,” covered by The Chipmunks
  • Part 29: “You’re The One That I Want,” from Grease
  • Part 30: “Only You,” by The Platters
  • Part 35: “Hound Dog,” by Elvis Presley

DeMorgan’s Laws:

  • Part 5: “Never Gonna Give You Up,” by Rick Astley
  • Part 28: “We’re Breaking Free,” from High School Musical (2006)

Simple nested predicates:

  • Part 6: “Everybody Loves Somebody Sometime,” by Dean Martin
  • Part 25: “Every Valley Shall Be Exalted,” from Handel’s Messiah
  • Part 33: “Heartache Tonight,” by The Eagles
  • Part 38: “Everybody Needs Somebody To Love,” by Wilson Pickett and covered in The Blues Brothers (1980)
  • Part 46: “Mean,” by Taylor Swift
  • Part 56: “Turn! Turn! Turn!” by The Byrds

Maximum or minimum of a function:

  • Part 12: “For the First Time in Forever,” by Kristen Bell and Idina Menzel and from Frozen (2013)
  • Part 19: “Tennessee Christmas,” by Amy Grant
  • Part 22: “The Most Wonderful Time of the Year,” by Andy Williams
  • Part 48: “I Got The Boy,” by Jana Kramer
  • Part 60: “I Loved Her First,” by Heartland

Somewhat complicated examples:

  • Part 11 : “Friends in Low Places,” by Garth Brooks
  • Part 27 : “There is a Castle on a Cloud,” from Les Miserables
  • Part 41: Winston Churchill
  • Part 44: Casablanca (1942)
  • Part 51: “Everybody Wants to Rule the World,” by Tears For Fears
  • Part 58: “Fifteen,” by Taylor Swift
  • Part 59: “We Are Never Ever Getting Back Together,” by Taylor Swift
  • Part 61: “Style,” by Taylor Swift

Fairly complicated examples:

  • Part 17 : Richard Nixon
  • Part 47: “Homegrown,” by Zac Brown Band
  • Part 52: “If Ever You’re In My Arms Again,” by Peabo Bryson

Really complicated examples:

  • Part 18: “Sleigh Ride,” covered by Pentatonix
  • Part 26: “All the Gold in California,” by the Gatlin Brothers
  • Part 40: “One of These Things Is Not Like the Others,” from Sesame Street
  • Part 42: “Take It Easy,” by The Eagles

Predicate Logic and Popular Culture (Part 61): Taylor Swift

Let S(t) be the proposition “We are in style at time t,” let C(t) be the proposition “We crash down at time t,” and let B(t) be the proposition “We come back at time t.” Translate the logical statement

\forall t (\lnot S(t)) \Rightarrow (\forall t(C(t) \Rightarrow \exists u>t(B(u))).

The straightforward way of translating this into English is, “If we never go out of style, then whenever we crash down we come back at a later time. This approximately matches the second half of the chorus of one of Taylor Swift’s hit songs.

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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.