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356538 (4) [Avatar] Offline
#1
Ok, I realize this is pretty minor, but your proof (on page 4) of two odd numbers always summing to an even number is circular. Do you see why?

"For any integer n, the formula 2n + 1 produces an odd number".

Kind of uses the hypothesis as a given, doesn't it? And later on "This is an even number because 2 times anything is even".

Anyway.... sorry to be petty here, I'm enjoying the book so far, really!
262479 (1) [Avatar] Offline
#2
Not sure that this is circular logic. You could rewrite that statement as "A whole number k is defined to be odd if there is some whole number n such that k = 2n + 1". That is an axiom, it is not derived from the hypothesis that "two odd numbers sum to an even number".
356538 (4) [Avatar] Offline
#3
Yes, you could rewrite the proof by first defining:

* an even integer as any integer that is divisible by two and

* an odd number as any even integer plus one.

Good to define these things formally before proving things about them.