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The book 'How to Read and Do Proofs' by Daniel Solow has the following exercise:

Find the contrapositive of the proposition, “If $n$ is an integer for which $n^2$ is even, then $n$ is even.”

I thought the answer would be: "If $n$ is not even, $n$ is not an integer or $n^2$ is not even"

(I reasoned that the condition in the exercise statement is an 'and' statement and hence it's negation involving 'or', according to De Morgan's law)

But the solution to this is given as: "If $n$ is an odd integer, then $n^2$ is odd"

I have these doubts:

  1. Why is my answer wrong, if it is?
  2. Is my negation of the condition, and the reasoning behind it, right?
  3. How is not("$n$ is even") equivalent to "$n$ is odd". Aren't there numbers which are neither odd nor even?
  4. Let $A$ be any statement. Can any statement $B$ which is logically equivalent to $\mathrm{not}(A)$ be called the negation of $A$?
Robin
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Boay
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  • It looks to me like the question is asking about statements over the natural numbers or integers, hence why your answer is different due to also containing the "$n$ is not an integer" part. As for the third part of your question - every integer is either odd or even. – Robin Jan 03 '24 at 12:48
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  • @Robin Are you saying that we must assume that n must be always treated as an integer in this question? It is not mentioned in the exercise and this question is not set in any context. (But I understand that mine and the author's solutions become logically equivalent). Also,please be kind and answer my 4th doubt if you can. – Boay Jan 03 '24 at 14:10
  • The book is using informal language. This makes a mechanical conversion to the contrapositive tricky. Even though the "$n$ is an integer" phrase was included in the if-then statement, the author meant it as an introduction of the variable $n$, establishing its domain, not as a clause of the implication. Thus the author only negated the "$n\text{/}n^2$ is even" parts of the statement. Yes, this is a bit confusing. That is why mathematicians generally prefer to use formal language. – Paul Sinclair Jan 04 '24 at 17:15
  • However, understanding formal language takes training, so using it limits your audience. Introductory material is thus often expressed informally so that it will reach neophytes, and lay the foundations of that training. And when informal language is used, meaning is generally more important than technical details, so yes, it is common to use logically equivalent statements to make the phrasing more natural. Your answer is technically correct, but the author's answer is also correct in the context in which it is used. Your take-away should be to recognize informal vs formal language. – Paul Sinclair Jan 04 '24 at 17:21
  • @Paul Sinclair, Thank you very much for your answer. It was very clear. But I want to ask something unrelated to the my question . At what level of mathematical exposure would you recommend reading a formal mathematical logic book (Like that of Enderton)? I am an absolute beginner in college maths.(Undergraduate freshman in physics interested in maths and currently reading a real analysis book). – Boay Jan 05 '24 at 12:08
  • Sounds very familiar, except that I was a junior when I took a first course in real analysis and discovered to my surprise that I had the soul of a mathematician. There are levels of formal language. True formalism is terse and difficult to follow. Very few mathematicians have much to do with it. It is worth some investigation, though, as it is how mathematics was freed from the shackles of "but which is the real theory?" (Answer: as long as a theory doesn't contradict itself, it is real, even if it contradicts other theories. They are just applicable in different places.) – Paul Sinclair Jan 05 '24 at 13:03
  • Before investigating such foundational works too deeply, I suggest mastering that initial real analysis text. For me, Real Analysis was my introduction to true, bottom-up, mathematics, where everything was precisely defined, and all proofs were rigorous. This is the level of formality where most mathematics is done. Some abstract algebra may also be useful in drilling these skills. Once you've mastered this rigorous approach, the whole world of math will be open to you to explore as you will. Part of that world is the study of foundations. – Paul Sinclair Jan 05 '24 at 13:11
  • @Paul Sinclair, Thank you for the advice. Much appreciated. – Boay Jan 05 '24 at 14:42

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