Morning Mathematical Monsters & Maniacs!
(Today’s post is sponsored by the letter “M”)
Over the past 600+ episodes, The Simpsons has taken us on an amazing mathematical journey involving fractions, probability, Fermat’s last theorem, and hundreds of other aspects from the wonderful world off mathematics.
And what better way to start your week, then by discussing math Monday morning?
This week we go back to an episode of The Simpsons that aired twenty years ago this week on December 19th 1999 and brought Venn diagrams into The Simpsons.
In Grift of the Magi (Season 11, Episode 09), an all-new staff at the elementary school do no actual teaching and instead teach the children about toy design and marketing. This leads to Lisa doing work on her own:
Teacher: Lisa, are you doing math?!
Lisa: Just a few Venn diagrams.
Ralph: There’s more under her chair.
So let’s look at some Venn diagrams.
A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. These diagrams depict elements as points in the plane, and sets as regions inside closed curves. A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set.
John Venn (4 August 1834 – 4 April 1923) was an English mathematician famous for introducing the Venn diagram in 1881. Venn diagrams are used in the fields of set theory, probability, logic, statistics, competition math, and computer science. Already famous for his 1866 publication The Logic of Chance, Venn then further developed George Boole’s theories in the 1881 work Symbolic Logic, where he highlighted what would become known as Venn diagrams. (English mathematician George Boole, 2 November 1815 – 8 December 1864, you may recall from Boolean algebra).
Here is a Venn diagram explaining the differences between Dweebs, Dorks, Geeks, and Nerds:
From the Venn diagram above, we can see…
Dweebs have intelligence and social ineptitude, without obsession;
Dorks have social ineptitude and obsession, without intelligence;
Geeks have obsession and intelligence, without social ineptitude;
Nerds have intelligence and social ineptitude and obsession.
Intersection of two sets A and B, denoted by A ∩ B, is the set containing all elements of A that also belong to B, but nothing else.
Union of two sets A and B, denoted by A ∪ B, is the set of elements which are in A, in B, or in both A and B.
Disjunctive Union of two sets A and B, denoted by A Δ B, is the set of elements which are in either set A or set B, and not in their intersection belonging to A and B.
Relative Complement of two sets A and B with A in B, also termed the set difference of B and A, is the set of elements in B but not in A.
Absolute Complement of a set A is any element in the universe (depicted by the letter U) that is not is the set A.
Here is a graphical depiction of these 5 basic operations:
So let’s go to another classic episode of The Simpsons to apply our knowledge of Venn diagrams and basic operations. In Dog of Death (Season 03, Episode 19), Springfield is “in the grip of lottery fever” with a $130 million jackpot.
The winning numbers of the $130 million jackpot are 3, 17, 26, 38, 41, and 49 (which Kent Brockman wins). The following week, the numbers are 3, 6, 17, 18, 22 and 29 (which Marge would have won if the Simpsons family weren’t cutting corners to make ends meet). Here is a Venn diagram depicting the lottery numbers:
In the circle to left, we see Kent’s 6 number. In the circle to the right, we see Marge’s 6 numbers. We see 2 numbers overlap and belong to both Kent and Marge. The numbers outside of the two circles are all the other numbers in the universe of this lottery that could have been selected by Kent or Marge, but weren’t.
So let’s see that in basic operations:
Intersection of Kent’s numbers and Marge’s numbers (denoted by Kent ∩ Marge) are 3 and 17 (numbers in blue boxes only).
Union of Kent’s numbers and Marge’s numbers (denoted by Kent ∪ Marge) are 3, 6, 17, 18, 22, 26, 29, 38, 41, and 49 (numbers in blue, red, and green boxes).
Disjunctive Union of Kent’s numbers and Marge’s numbers (denoted by Kent Δ Marge) are 6, 18, 22, 26, 29, 38, 41, and 49 (numbers in red and green boxes; not blue).
Relative Complement of Kent’s numbers in Marge’s numbers are 6, 18, 22, and 29 (numbers in green boxes only; not red or blue).
Relative Complement of Marge’s numbers in Kent’s numbers are 26, 38, 41, and 49 (numbers in red boxes only, not green or blue).
Absolute Complement of Kent’s numbers are all 77 numbers that Kent did not select (numbers in green boxes and all numbers in white not selected; not red or blue).
Absolute Complement of Marge’s numbers are all 77 numbers that Marge did not select (numbers in red boxes and all numbers in white not selected; not green or blue).
Now that we’ve completed a look at Venn diagrams, why not show your love for math and numbers and sets with your own Math Mayhem shirt or hoodie.
If you love math or enjoy reading these posts, don’t forget to stop by the Addicts Shop and check out all the paraphernalia, including the Math Mayhem shirts and hoodies.
Were you familiar with Venn diagrams? What’s your favourite Venn diagram? Do you recall the episode featuring Venn diagrams from 20 years ago? Did you recall the episode with lottery numbers? Are you a dweek, dork, geek, nerd, or none of the above? Sound off in the comments below. You know we love hearing from you.