"What is the cardinality of the complement of set A?"
I choose to blog this problem not because it's a Geometry question, but because it's a logic question related to what we're currently reading in Eugenia Cheng's book. This question even contains a Venn diagram similar to Cheng's -- the universal set U is the set of whole numbers less than 26, while A is the set of all numbers less than or equal to 0.
So far, this is a strange question. The set of whole numbers less than 26 is {0, 1, 2, 3, 4, ..., 25}. But A can only be the set {0} -- even though there are many real numbers less than or equal to 0, there are no whole numbers less than 0. In a given Venn diagram, every object mentioned in the problem must be an element of the universal set (hence the name "universal"). No object in U is less than or equal to 0 besides 0, and so A = {0}. If an object is in A, then it is in U -- A is a subset of U.
This leaves {1, 2, 3, 4, ..., 25} as the complement of A -- as Cheng would put it in her Chapter 4, it represents the negation of A. So far, she doesn't write about cardinality in her third book, but she does discuss it in her second book, Beyond Infinity. The cardinality of a set is basically the number of elements that it contains.
The set U \ A = {1, 2, 3, 4, ..., 25} clearly has 25 elements. Therefore the desired cardinality is 25 -- and of course, today's date is the 25th.
It's nice that I was able to use Cheng's books to solve today's Pappas problem. But now it's time to proceed with the next chapter.
Chapter 7 of Eugenia Cheng's The Art of Logic in an Illogical World is called "How to Be Right." It begins as follows:
"In the lovely stop motion film Chicken Run, the smooth-talking American rooster Rocky does a deal with the sly salesmen rats saying he will play them 'all the eggs he lays that month.' The rats are sly but not very knowledgeable about chickens, so they don't realize that roosters don't lay eggs, and that the total number of eggs Rocky lays that month is therefore going to be zero."
This chapter is "all" about quantifiers -- those little words like "all." Cheng tells us that technically, Rocky isn't lying -- he does give the rats "all" zero of his eggs.
Here is Cheng's next example:
"However, to express that fed-up-ness it may be tempting to say 'All men are sexist pigs!' but this might provoke someone to argue that not every man is a sexist pig, say, perhaps, Justin Trudeau [the Canadian prime minister]. In this case you've expressed true emotions but with inaccurate logic, and in doing so you're tempted certain types of people to argue with your logic instead of soothe your emotions."
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next two weeks and skip all posts that have the "Eugenia Cheng" label.
The author explains that this is how we find the negation of statements quantified by "all" (or another related word):
Statement: You never do the washing up!
Negation: I did the washing up one time.
Statement: You always leave a mess in the kitchen.
Negation: There was one time I did not leave a mess in the kitchen.
But as Cheng points out, people are prone to making sweeping statements. In fact, that last sentence is itself a sweeping statement. Is everyone prone to making sweeping statements? Cheng answers:
"I think everyone I know is, but I've met only a tiny proportion of people, so really I should just say 'Everyone I know is prone to making sweeping statements.' I have now refined my statement and made it less ambiguous, hence more defensible using logic."
Cheng tells us, though:
"Looking for the truth in someone's statement can be much more productive than pedantically demonstrating that they are wrong. I think this is an instance of the principle of charity, where you try to think the best of everyone all the time."
According to the author, there are two logically unambiguous ways to qualify a statement:
- The statement is true of everything in your world. Perhaps "all mathematicians are awkward."
- The statement is true of at least one thing in your world. Perhaps "there is at least one mathematician who is friendly." (Cheng hopes she counts as this. Don't worry, Eugenia -- I, at least, think you're friendly. Thus there exists at least one blogger who thinks you're friendly.)
Compare these two statements:
Everyone in the US is obese.
Someone in the US is obese.
Formally these two types of statement would be rendered using "for all" and "there exists" like this:
For all people X in the US, X is obese.
There exists a person X in the US such that X is obese.
As Cheng explains:
"'For all' and 'there exists' are called quantifiers in mathematics; they quantify the scope of our statement."
Now consider this statement:
All the elephants in the room have two heads.
According to the author, this statement is vacuously true. She explains why:
"There are no elephants in the room, and all zero of them have two heads. This is related to the fact that a falsehood implies anything, logically."
Over the summer, I attempted to sneak spherical geometry into high school Euclidean geometry by taking advantage of vacuous truth. For example, the statement "if two parallel lines are cut by a transversal, then corresponding angles are congruent" is obviously true in Euclidean geometry -- and it's vacuously satisfied in spherical geometry as well, because there are no parallel lines! Since then, I decided that this example is confusing rather than illuminating, and so I no longer recommend this as something to teach high school students.
Let's return to Cheng. Her earlier example "All men are sexist pigs!" can be rewritten as:
For all X in the set of men, X is a sexist pig.
And this is the negation of the above statement:
There exists X in the set of men, such that X is not a sexist pig.
According to the author, this statement is true, because her friend Greg is not a sexist pig. So the original statement must be false.
Cheng now describes one of her favorite mathematical jokes:
Three logicians walk into a bar. The bartender says "Would everyone like a beer?" The first logician says "I don't know." The second logician says "I don't know." The third logician says "Yes."
Let's explain the joke. Either:
Everyone in the US is obese.
Someone in the US is obese.
Formally these two types of statement would be rendered using "for all" and "there exists" like this:
For all people X in the US, X is obese.
There exists a person X in the US such that X is obese.
As Cheng explains:
"'For all' and 'there exists' are called quantifiers in mathematics; they quantify the scope of our statement."
Now consider this statement:
All the elephants in the room have two heads.
According to the author, this statement is vacuously true. She explains why:
"There are no elephants in the room, and all zero of them have two heads. This is related to the fact that a falsehood implies anything, logically."
Over the summer, I attempted to sneak spherical geometry into high school Euclidean geometry by taking advantage of vacuous truth. For example, the statement "if two parallel lines are cut by a transversal, then corresponding angles are congruent" is obviously true in Euclidean geometry -- and it's vacuously satisfied in spherical geometry as well, because there are no parallel lines! Since then, I decided that this example is confusing rather than illuminating, and so I no longer recommend this as something to teach high school students.
Let's return to Cheng. Her earlier example "All men are sexist pigs!" can be rewritten as:
For all X in the set of men, X is a sexist pig.
And this is the negation of the above statement:
There exists X in the set of men, such that X is not a sexist pig.
According to the author, this statement is true, because her friend Greg is not a sexist pig. So the original statement must be false.
Cheng now describes one of her favorite mathematical jokes:
Three logicians walk into a bar. The bartender says "Would everyone like a beer?" The first logician says "I don't know." The second logician says "I don't know." The third logician says "Yes."
Let's explain the joke. Either:
- A: All three logicians want a beer.
- not A: There exists a logician who does not want a beer.
The first logician wants a beer. That's not enough to conclude A (since we don't know what the other two logicians want), but there's no logician for whom we can conclude not A either.
The second logician wants a beer. That's not enough to conclude A (since we don't know what the other third logician want), but there's still no logician for whom we can conclude not A either.
The third logician wants a beer. Finally, that's enough to conclude A, and so the answer is "Yes."
Here is Cheng's next example -- "all mathematicians are awkward." That is:
For all X in the set of mathematicians, X is awkward.
But Cheng doesn't consider herself to be awkward. Written as an implication, this statement becomes:
Being a mathematician implies being awkward.
In other words, either Cheng is awkward, or she isn't a mathematician. She tells us that she takes offense at both of these claims.
Cheng next writes, "Every female science student has been hit on by their supervisor." This is easy to refute -- a counterexample is any female science who has never been hit on by their supervisor. The author tells us that she herself is a counterexample. She proceeds:
"Whereas if I say 'Some female science students have been hit on by their supervisor' and you want to say that this isn't true, you have to do something much harder -- you have to check every single female science student, and make sure that nobody has been hit on by their supervisor. Unfortunately this won't be possible."
We have the following negations:
World: All female science students in time.
Original statement: For all female science students X, X has been hit on by their supervisor.
Negation: There exists a female science student X such that X was not hit on by their supervisor.
World: All female science students in time.
Original statement: There exists a female science student X such that X was hit on by their supervisor.
Negation: For all female science students X, X has not been hit on by their supervisor.
Cheng tells us that by using quantifiers, we now have predicate logic, or first-order logic. The opposite of predicate logic is propositional logic (no quantifiers). She explains "first-order":
"'First-order' is to distinguish it from higher-order versions of logic, which are more complicated in the way that the quantifiers work."
The author tells us that if we want to be right more often, we can be more careful with sweeping generalizations by using the following phrases:
In my opinion...
In my experience...
Maybe...
Sometimes...
Apparently...
It seems to me...
Cheng's PhD supervisor, Martin Hyland, often uses the phrase "There is a sense in which":
"'There is a sense in which Mozart is more boring than Brahms' is another way of correcting my sweeping statement 'Mozart is more boring than Brahms.' 'There is a sense in which teaching math can be a waste of time and money.'"
The author tells us that the last statement above could have been helpful earlier when she was in a Twitter debate with a traditionalist.
Cheng concludes the chapter with some advice:
"Someone might say something that is untrue in strictly logical terms, but perhaps they were really trying to say something else, perhaps something with strong emotional content that we should listen to if we are intelligent humans rather than intelligent emotionless robots."
This is what I wrote last year about today's lesson:
Well for today, we intelligent humans are looking at our second day of the Chapter 2 review. I've decided that when we have these two-day reviews, I'll use the second day to post interesting activities that I find from other sources.
My usual go-to sites, Fawn Nguyen and Sarah Carter, are out for Geometry, since the former is a middle school teacher and the latter is an Algebra II teacher. But hey -- the most well-known Geometry site is probably now Shaun Carter, ever since he married Sarah. (That's right -- just as physics nerd Sheldon is married to a fellow nerd, Shaun Carter married a fellow math teacher.)
The following worksheet is all about the three undefined terms (point, line, plane). Recall that the first question on the test asks students to identify these undefined terms. The page didn't print well for me, so you might want to find this worksheet at the original source:
https://blog.primefactorisation.com/2017/09/09/undefined-terms-inb-pages/
My usual go-to sites, Fawn Nguyen and Sarah Carter, are out for Geometry, since the former is a middle school teacher and the latter is an Algebra II teacher. But hey -- the most well-known Geometry site is probably now Shaun Carter, ever since he married Sarah. (That's right -- just as physics nerd Sheldon is married to a fellow nerd, Shaun Carter married a fellow math teacher.)
The following worksheet is all about the three undefined terms (point, line, plane). Recall that the first question on the test asks students to identify these undefined terms. The page didn't print well for me, so you might want to find this worksheet at the original source:
https://blog.primefactorisation.com/2017/09/09/undefined-terms-inb-pages/
By the way, Shaun, like his wife Sarah, has left their old school. In this post, he explains that he is now pursuing a masters degree:
He says that he's been awarded a "teaching assistantship," which means that he'll likely be teaching undergrads, not high school students. Thus he's not currently teaching Geometry. Still, I post his worksheet from last year:
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