Gaining Approval: Insights From “How To Prove It”
[Note: These are insights that surprised/connected the dots for me personally. I feel dumb admitting some of these, but my hope is that this’ll be a good data point for others]
How to write a math book:
1. It’s fun to prove things wrong. Examples such as “What’s wrong with this proof” really worked for me.
2. A great pattern for teaching: a solid explanation of new concept, 3-5 example problems to work through, and solutions & commentary immediately after! It had a short feedback loop that I felt myself wanting to be a part of.
3. Concepts built upon each other in quick succession, so you don’t have to flip back and review.
Proofs:
1. Proofs aren’t made to show how they made the proof, just what they’re proving in a succinct way. Since this is the case, it’s okay to be confused even if the proof sounds like everything of course follows everything else, and you’re an idiot for not immediately following. The proof took hours/weeks/years/centuries to condense in that form, so it’s reasonable if it takes a few hours & math stack exchange to understand it.
2. There are tips and tricks for writing proofs depending on the properties of what you’re proving. For instance, it’s generally easier to prove a positive, so if you’re proving a negative, make it a positive and do a proof by contradiction on it!
Math:
1. A ⟷ B is making two claims: A ⟶ B & A ⟵ B.
2. Intuition for material implication (rule of inference):
Either Bob did it or Alice did it. If it’s not Bob, then it’s Alice.
It’s either A or B. If it’s not A, then it’s B.
A ∨ B ⟷ ¬A ⟶ B
3. Intuition for contrapositive of implication:
¬A ⟶ B ⟷ A ∨ B (rule of inference above)
A ∨ B ⟷ B ∨ A (Bob or Alice = Alice or Bob)
B ∨ A ⟷ ¬B ⟶ A (rule of inference above)
¬B ⟶ A ⟷ ¬A ⟶ B (Bam! Contrapositive!)
4. Combining induction with functions gets you recursion.
5. X = Ø is a negative statement: ∀y(y ∉ X)
6. X is infinite is a negative statement: not finite.
Math, Dancing, and Music:
I improv dance & piano, and to hit on something interesting, you have to combine different relationships with different objects. Let’s hone in on just negation.
In math, negate “a subset of” to get “disjoint”, “intersection of” to get “symmetric difference”, “for all x, P(x)” to get “there exist an x where ¬P(x)” (plus the 3-4 mentioned above).
In dance, you can double your repertoire by simply applying negation to all of your dance moves. Even reversing walking or turning your head looks smooth (results may vary). The only proper name dance reversals I can think of is the running man & the jerk.
In music, reversing your melody, chord progression order, or meter can also produce interesting results. Example: play heart-and-soul’s melody backwards. Chords to go with it: Amin, F, Dmin, Emin (though you will have to shift a couple notes if you use those chords to make it sound right).
...that’s just negation, too! This idea reminds me of alkjash’s Hammer and Nails with negation being the hammer and the above mentioned being the nails. One possible implication of this is that any new relationship you learn in math, dance, music, etc. could be applied to all objects/properties you have in math, dance, music, etc. to produce something interesting. I would like to explore this idea in a much greater depth in the future.
Title and review inspired by TurnTrout
But this is not true, because ¬(¬A ⟶ B) ⟶ A ∨ B. With what you’ve written you can get from the left side to the right side, but you can’t get from the right side to the left side.
What you need is: “Either Alice did it or Bob did it. If it wasn’t Alice, then it was Bob; and if it wasn’t Bob, then it was Alice.”
Thus: A ∨ B ⟷ (¬A ⟶ B ∧ ¬B ⟶ A)
No, what he’s written is correct[0]. A ∨ B, ¬A ⟶ B, and ¬B ⟶ A are all equivalent. (Hence your last statement is also correct!) Note for instance that the last two are just contrapositives of one another and so equivalent.
[0]In classical logic, obviously, for the nitpickers; that’s all I’m going to consider here.
Ah, of course. Thanks.
Note also you could easily see your initial comment to be wrong just by computing the truth tables! Equivalence in classical propositional logic is pretty easy—you don’t need to think about proofs, just write down the truth tables!