I must once again resort to the advice of this great community.
As I was reading about the pigeonhole principle something about its proof struck me as odd. Allow me to explain:
After reading the "The Foundations: Logic and Proofs" chapter in Rosen's "Discrete mathematics and its applications" book I was left with the feeling/notion that I can (and I ought to) describe all my proof's statements in symbols.
Yet, as you can see in the pigeonhole principle's proof:
We use a proof by contraposition. Suppose none of the k boxes has more than one object. Then the total number of objects would be at most k. This contradicts the statement that we have k + 1 objects.
Without a doubt it has more English words than symbols. Yet the proof is actually without flaws. I struggle with the fact that I can't convert it into the A -> B format (I hope you understand what I'm trying to convey).
However, is there a way to symbolically represent what it states (as I'm trying to put it in my mind)?
Or the only way to argue for this proof, is by using words?
And if so, is there any guide or principle that should tell us when to use words or symbols in our proofs?