Category: Discrete mathematics

Predicate Logic and Popular Culture (Part 127): Hamilton

Let L(x) be the proposition “x is legal in New Jersey.” Translate the logical statement

\forall x L(x).

This translates as “Everything is legal in New Jersey,” one of the running gags in the hit musical Hamilton.

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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 126): Hamilton

Let W(x,t) be the proposition “You walk with x at time t.” Translate the logical statement

\lnot \exists t \forall x \lnot W(x,t).

The straightforward way of translating this into English is, “If you stand for nothing, then you’ll fall for anything,” one of the motifs (with a slightly different wording) of the hit musical Hamilton.

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Context: I recently 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 125): INXS

Let M(x) be the proposition “x is a man,” let W(x) be the proposition “x is with you,” and let K(x) be the proposition “I knew x.” Translate the logical statement

\forall x ((M(x) \land W(x)) \Rightarrow K(x)).

This matches the chorus of “Do Wot You Do” by INXS, which also was on one of the all-time best movie soundtracks.

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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 124): High School Musical

Let W(x) be the proposition “I want to do x.” Translate the logical statement

W(\hbox{be with you}) \land \forall x ((x \ne \hbox{be with you}) \Rightarrow \lnot W(x)).

This matches the chorus of one of the songs from High School Musical 3.

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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: 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. My experiences teaching these ideas to my discrete mathematics students led to my recent publication (John Quintanilla, “Name That Tune: Teaching Predicate Logic with Popular Culture,” MAA Focus, Vol. 36, No. 4, pp. 27-28, August/September 2016).

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: “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
  • Part 98: “Call Me Maybe,” by Carly Rae Jepsen.

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
  • Part 76: “You’re Never Fully Dressed Without A Smile,” from Annie.
  • Part 109: “Dancing in the Dark,” by Bruce Springsteen.
  • Part 122: “Keep Yourself Alive,” by Queen.

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)
  • Part 65: “Stars and Stripes Forever,” by John Philip Sousa.
  • Part 68: “Love Yourself,” by Justin Bieber.
  • Part 69: “I Will Always Love You,” by Dolly Parton (covered by Whitney Houston).
  • Part 74: “Faithfully,” by Journey.
  • Part 79: “We’re Not Gonna Take It Anymore,” by Twisted Sister.
  • Part 87: “Hungry Heart,” by Bruce Springsteen.
  • Part 99: “It’s the End of the World,” by R.E.M.
  • Part 100: “Hold the Line,” by Toto.
  • Part 101: “Break My Stride,” by Matthew Wilder.
  • Part 102: “Try Everything,” by Shakira.
  • Part 108: “BO$$,” by Fifth Harmony.
  • Part 113: “Sweet Caroline,” by Neil Diamond.
  • Part 114: “You Know Nothing, Jon Snow,” from Game of Thrones.
  • Part 118: “The Lazy Song,” by Bruno Mars.
  • Part 120: “Cold,” by Crossfade.
  • Part 123: “Always on My Mind,” by Willie Nelson.

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)
  • Part 62: “All My Exes Live In Texas,” by George Strait.
  • Part 70: “Wannabe,” by the Spice Girls.
  • Part 72: “You Shook Me All Night Long,” by AC/DC.
  • Part 81: “Ascot Gavotte,” from My Fair Lady
  • Part 82: “Sharp Dressed Man,” by ZZ Top.
  • Part 86: “I Could Have Danced All Night,” from My Fair Lady.
  • Part 95: “Every Breath You Take,” by The Police.
  • Part 96: “Only the Lonely,” by Roy Orbison.
  • Part 97: “I Still Haven’t Found What I’m Looking For,” by U2.
  • Part 105: “Every Rose Has Its Thorn,” by Poison.
  • Part 107: “Party in the U.S.A.,” by Miley Cyrus.
  • Part 112: “Winners Aren’t Losers,” by Donald J. Trump and Jimmy Kimmel.
  • Part 115: “Every Time We Touch,” by Cascada.
  • Part 117: “Stronger,” by Kelly Clarkson.

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)
  • Part 93: “There’s No Business Like Show Business,” from Annie Get Your Gun (1946).
  • Part 94: “Not While I’m Around,” from Sweeney Todd (1979).
  • Part 104: “Wild Blue Yonder” (US Air Force)
  • Part 106: “No One,” by Alicia Keys.
  • Part 116: “Ocean Front Property,” by George Strait.

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
  • Part 73: “Dust In The Wind,” by Kansas.
  • Part 75: “Happy Together,” by The Turtles.
  • Part 77: “All She Wants To Do Is Dance,” by Don Henley.
  • Part 90: “All You Need Is Love,” by The Beatles.

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
  • Part 63: P. T. Barnum.
  • Part 64: Abraham Lincoln.
  • Part 66: “Somewhere,” from West Side Story.
  • Part 71: “Hold On,” by Wilson Philips.
  • Part 80: Liverpool FC.
  • Part 84: “If You Leave,” by OMD.
  • Part 103: “The Caisson Song” (US Army).
  • Part 111: “Always Something There To Remind Me,” by Naked Eyes.
  • Part 121: “All the Right Moves,” by OneRepublic.

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
  • Part 92: “Anything You Can Do,” from Annie Get Your Gun.
  • Part 119: “Uptown Girl,” by Billy Joel.

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
  • Part 67: “When I Think Of You,” by Janet Jackson.
  • Part 78: “Nothing’s Gonna Stop Us Now,” by Starship.
  • Part 89: “No One Is Alone,” from Into The Woods.
  • Part 110: “Everybody Loves My Baby,” by Louis Armstrong.

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
  • Part 83: “Something Good,” from The Sound of Music.
  • Part 85: “Joy To The World,” by Three Dog Night.
  • Part 88: “Like A Rolling Stone,” by Bob Dylan.
  • Part 91: “Into the Fire,” from The Scarlet Pimpernel.

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

Solving a Math Competition Problem: Part 9

This series of posts concerns solving the following problem from the 2016 University of Maryland High School Mathematics Competition.

A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?

a. 2^8

b. 2^9

c. 81

d. 76

e. 74

This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

Reflection
I didn’t really need the projection into the plane for the solution, but my problem-solving self needed it to be able to count points and regions in slow motion. So, I should present a cleaned-up solution:

 

Solution
Since there are 9 planes, each plane must intersect with every other in a line, creating two points on the surface of the sphere. Thus, there are (9∗8)/2 * 2 = 72 points of intersection, and for n planes, there are 𝑛(𝑛 − 1) points of intersection. With the first plane, there are zero points of intersection and two regions. Suppose we now have n planes and N regions. We add another plane, creating a circle on the sphere. For each segment that the circle intersects, it creates an additional intersection point as it enters, and it divides the region into two parts, adding one additional region. Hence, for each point added, a region is added as well. Since there are two
regions with zero points, there are thus 74 regions with 72 points of intersection.

Solving a Math Competition Problem: Part 8

This series of posts concerns solving the following problem from the 2016 University of Maryland High School Mathematics Competition.

A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?

a. 2^8

b. 2^9

c. 81

d. 76

e. 74

This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane.

The projection of four planes:

Conjecture: The max number of regions is the number of intersection points plus 2.
Proof (by induction)
If we have 1 plane, we have no intersection points and 2 regions. Suppose we have n planes with 𝑛(𝑛 − 1) intersection points and 𝑛(𝑛 − 1) + 2 regions. Now we add the next plane to our figure. The plane creates a circle on the sphere. To maximize the number of regions, we angle the plane so that our circle does not intersect any already-existing intersection points. So the circle goes through a number of segments. Each time it does, it cuts the region bounded by that segment into two. So for each new intersection point, we lose one region and gain two, for a net gain of one region. That is, however many intersection points are added, that will be the number of regions added as well. And since 𝑛 + 1 planes have (𝑛 + 1)(𝑛) intersection points, we will
have (𝑛 + 1)(𝑛) + 2 max regions. DONE.
For the original competition problem, we have 9 planes and hence 9*8 + 2 = 74 regions, answer e.

Solving a Math Competition Problem: Part 7

This series of posts concerns solving the following problem from the 2016 University of Maryland High School Mathematics Competition.

A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?

a. 2^8

b. 2^9

c. 81

d. 76

e. 74

This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane.

The projection of four planes:

After a while, I had a chart for max possible regions.

  • 1 plane: Max regions = 2
  • 2 planes: Max regions = 4
  • 3 planes: Max regions = 8 (exponential?)
  • 4 planes: Max regions = 14 (nope!)
  • 5 planes: Max regions = 22 (huh?)

Then, really because I had no other ideas, I tried counting intersection points AND max regions
(remembering that one intersection point is “at infinity” – that is, the north pole).

  • 1 plane: Intersection Points = 0, Max regions = 2
  • 2 planes: Intersection Points = 2, Max regions = 4
  • 3 planes: Intersection Points = 6, Max regions = 8
  • 4 planes: Intersection Points = 12, Max regions = 14
  • 5 planes: Intersection Points  20, Max regions = 22

Oh. My. Goodness. The max regions are simply the number of intersection points plus 2. Could it really REALLY be that simple?

Solving a Math Competition Problem: Part 6

This series of posts concerns solving the following problem from the 2016 University of Maryland High School Mathematics Competition.

A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?

a. 2^8

b. 2^9

c. 81

d. 76

e. 74

This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane.

The projection of four planes:

After a while, I had a chart for max possible regions.

  • 1 plane: Max regions = 2
  • 2 planes: Max regions = 4
  • 3 planes: Max regions = 8 (exponential?)
  • 4 planes: Max regions = 14 (nope!)
  • 5 planes: Max regions = 22 (huh?)

Solving a Math Competition Problem: Part 5

This series of posts concerns solving the following problem from the 2016 University of Maryland High School Mathematics Competition.

A sphere is divided into regions by 9 planes that are passing through its center. What is the largest possible number of regions that are created on its surface?

a. 2^8

b. 2^9

c. 81

d. 76

e. 74

This series was actually written by my friend Jeff Cagle, department head for mathematics at Chapelgate Christian Academy, as he tried technique after technique to solve this problem. I thought that his resolution to the problem was an excellent example of the process of mathematical problem-solving, and (with his permission) I am posting the process of his solution here. (For the record, I have no doubt that I would not have been able to solve this problem.)

OK, so I wanted to prove that each region would be a triangle. So I decided to project the sphere onto a plane.

For a while, I toyed with the situation where we have

  • Plane 1 – equator (this always happens: Just make plane 1 the equator) 𝑃1(0𝑁, 0𝐸).
  • Plane 2 – Prime Meridian 𝑃2(90𝑁, 0𝐸)
  • Plane 3 – Intl Date Line 𝑃3(90𝑁, 90𝐸)
  • Plane 4 – at an angle to all of those 𝑃4(45𝑁, 45𝐸)

Here is our mapping with P1, P2, and P3 on it:

Now, how to represent P4? Aha! The inside of the unit circle is the southern hemisphere, and the outside is the northern. P4 must hit the equator a two points 180 degrees apart, go inside the southern hemisphere, and then outside to the northern. Thus:

The white region is a NONtriangular region created by the intersection of four planes. These are strange-looking regions, and I spent a long time – several days – vainly trying to count max regions created when I added P5, P6 etc. But one thing was clear: not all of the regions are triangular, nor can they be. For if a plane (say P4) cuts through a triangular region, it will create a new triangular region and a non-triangular “quadrilateral”, as in the figure below. So counting triangles from points is NOT the solution here!