Error in page 149-150: Chapter 3.3, Exercises 15 and 18

[MetallicCrimson]

In exercises 13-19, all the solutions are valid, and for (b), I especially liked how there's a formal and informal negation for each of the statements. However, for (a), the book says "without using the symbol ∀ or ∃ or variables", so I believe the following lines are redundant:

From Exercise 15:

• For every odd integer n, there is an integer k such that n = 2k + 1.
• Given any odd integer n, we can find another integer k so that n = 2k + 1.
• Every odd integer has the form 2k + 1 for some integer k.

From Exercise 18:

• For every real number x, there is a real number y such that x+y = 0.
• Given any real number x, there exists a real number y such that x + y = 0.

This still leaves 2 forms in both ex. 15 and ex. 18, which I think is enough.

Exercises 13, 14, 16, 17 and 19 don't contain redundant lines.

(I'd also put a comma such that
In 13 − 19 there are other correct answers in addition to those shown.
becomes
In 13 − 19, there are other correct answers in addition to those shown.)

[spamegg1]

I tried to follow the solution in the back of the book and its format:

I am not sure why it's written redundantly like that, maybe to have as many interpretations as possible?

[MetallicCrimson]

Aaah, I see.

maybe to have as many interpretations as possible

That might be the case. It's still a mistake (from the book, then) in my eyes, but I understand that you were following that format. Do you think it's better to keep the extra lines?

[spamegg1]

No, let's keep it without the comma.

[MetallicCrimson]

So I wanted to finish the 3.4 before trying to fix the issues mentioned in the discussions (just to keep chapter 3 together), but since I haven't found anything wrong, I'm ready.

However, I have another suggestion.

I'd replace in ex. 15
Informal statement: Every odd integer equals twice some other integer plus 1.
with
Informal statement: Every odd integer equals twice some integer plus 1.
I know "other" doesn't necessarily mean "different", but it could be confusing to some people when they try to find another integer. And since -1 is equal twice itself plus 1 (and it is an odd integer), I think the version without "other" is clearer.

In the same spirit, I'd replace in ex. 18
Informal statement: Every real number can be added to some other real number (possibly itself) to obtain 0.
with
Informal statement: Every real number can be added to some real number to obtain 0.

Also in ex. 16
Informal statement: There is a real number such that for every other real number, the product of the first real number and the second real number are equal.
with
Informal statement: There is a real number such that for every real number, the product of the first real number and the second real number are equal.

[MetallicCrimson]

So I noticed a problem with my fixes here:

"multiplied with" would be better as "multiplied by", but that's not the major issue. I actually got my suggested sentence backwards (it says uv = u instead of uv = v), so the one I'd suggest instead is:
There is a real number which doesn't change any real number when multiplied into it.

(Sorry, seems like I wasn't paying enough attention.)

[spamegg1]

I don't like "doesn't change" and "into it". It's getting vague and confusing.

Honestly these are limitations of English / natural languages, they suck at describing math if you remove variables. (Not your fault)

We are starting to overthink things too much, let's just leave it as it is (your previous suggestion).

[MetallicCrimson]

Got it, I'll leave it that way.

Two more small typos in my suggestions (that I already fixed):
In ex. 17: For all rational number r, there exists two integers should be there exist two integers

In ex. 19: There exists a real number R such that for every real number y should be There exists a real number x such that for every real number y (whoops, variables)

[spamegg1]

Closed by #50 thanks!