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Before explaining my issue, I wanted to first explain what things do make sense to me.

So, statements (used at the beginning of proofs) like "Suppose $x$ is an integer" or "Assume $x$ is an integer" make sense to me. The way I read them is like: "Let's just pretend that the symbol $x$ happens to represent an integer".

Statement (also used in proofs) like "Let $x$ be 3" or "Let $x$ equal 3" also make sense to me. The way I read them is like: "Let's just temporarily name 3 with the symbol $x$.

My issue comes with statements like "Let $x$ be an integer" or "Let $x$ $\in$ $\mathbb{Z}$". I really do not know how I should intuitively interpret such a statement. I can't interpret it in the same way as like "Let $x$ be 3" because there is not a specific object being assigned like 3. Should I interpret it like how I interpret "Suppose $x$ is an integer"?


EDIT:

So, from what I gathered from the responses, I think I understand now how I should think of "Let $x$ be an integer".

I could think of it as "Assume a newly created symbol x happens to represent an integer". However, this can cause issues as then "Let $x$ be an element of the empty set" is also completely valid.

Instead, I should not think of "Let $x$ be an integer" as an assumption, but an assignment/declaration, just like "Let $x$ be 3". One tangible way to think about this is to imagine myself assigning the newly created symbol $x$ to an integer chosen in secret by a friend. With this mindset, $x$ is not assumed to be an integer - it is an integer. It is just unknown to me what integer it is.

I hope this made sense to anybody with similar questions. If anyone thinks I have made an incorrect finding, please feel free to correct me.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$
    – Xander Henderson
    Mar 10, 2021 at 3:41
  • $\begingroup$ In my mind, if $x$ has not been introduced yet, then "Let $x$ be an integer" means the same thing as "Suppose $x$ is an integer." If $x$ has already been introduced, then of course the latter phrase would suggest that we're either about to do proof by contradiction or proof by cases or something like that. $\endgroup$
    – littleO
    Sep 21, 2021 at 19:25
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    $\begingroup$ Unfortunately, none of the 3 existing answers by Paul Frost, ryang or Karl are really correct. Read this post to grasp the issue fully. $\endgroup$
    – user21820
    Oct 23, 2021 at 9:52

3 Answers 3

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  1. The verb ‘letassigns meaning (definitions or restrictions or values) to variables or phrases or symbols.

    • In “let $k$ be an integer and $x=2k$”, $x$ is an arbitrary even number (or: the representation or general specification of the even numbers).

    • In “let $x=7$”, $x$ has been instantiated with a specific value.

      (or: “Put $x=7.$”)

  2. The adjective ‘arbitrary’ (signalling universal quantification; ‘Any’ versus ‘arbitrary’) is frequently tacit, and these sentences all mean the same:

    • Let $x$ be an arbitrary real number and suppose that $P(x).$
    • Let $x$ be real and suppose that $P(x).$
    • Let $x\in\mathbb{R}$ and suppose that $P(x).$
    • Let $x$ be an arbitrary real number such that $P(x).$
    • Let $x$ be real such that $P(x).$
    • Let $x\in\mathbb{R}$ such that $P(x).$
  3. The verbs ‘suppose’ and ‘assume’ introduce conditions or (provisional) assumptions.

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  • $\begingroup$ Thank you Ryan for your reply. So is "Let x be an integer" saying something like "Let's start by creating a new symbol x. Now, let's assume x happens to represent an integer? $\endgroup$ Mar 8, 2021 at 2:20
  • $\begingroup$ So, is "Let x be an integer" sort of like assigning "x" to an integer that is chosen in secret? This would make "x" represent an unknown integer and then no assumption would be needed. $\endgroup$ Mar 8, 2021 at 2:24
  • $\begingroup$ Sure no problem. Here was my comment: @Ryan G This link was very helpful. One issue though is that the example given of a definition is pretty much akin to "Let x be 3" where x is being assigned to something specific, unlike "Let x be an integer". In the case of the link, the specific thing being assigned to N is "the set of natural numbers". $\endgroup$ Mar 8, 2021 at 2:27
  • $\begingroup$ Just to elaborate on the "chosen in secret", I imagine like having a friend choose an integer in secret, place it in a box, and I use "x" to denote the integer in the box (although I don't know what integer is in the box). If "Let x be an integer" is not an assumption, but an assignment/declaration (like in programming), then I think it has a similar feel to the box analogy. $\endgroup$ Mar 8, 2021 at 2:36
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    $\begingroup$ Thank you for your help Ryan. I will make a summary in my answer on what I learned from everybody's reponses. $\endgroup$ Mar 8, 2021 at 3:08
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"Let" is usually used to introduce a new symbol along with an assumption about it ("Let $G$ be a group") or when defining notation ("Let $\mathcal P(S)$ denote the power set of $S$"). "Suppose" and "assume" are more conventional when introducing new assumptions using only existing symbols and notation ("Suppose $x^2<5$"). This isn't a rule, though - you can certainly introduce new symbols with an "assume" or "suppose".

I'd also say "suppose" is slightly more common when introducing a proposition that is "in doubt" and will shortly be refuted or discarded (e.g. with a proof by contradiction), whereas "assume" is more neutral in tone and is more common when introducing long-lasting assumptions or "background information".

The three words have the same technical meaning, in that we'd translate them into formal logic in the same way.

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  • $\begingroup$ So "Let x be an integer" pretty much says "Suppose x is an integer" without the implication that "x" can be an integer is in doubt? Like, for example, you wouldn't say "Let x be an element of the empty set", but you would say "Suppose x is an element of the empty set"? $\endgroup$ Mar 8, 2021 at 1:26
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    $\begingroup$ Yeah, that's roughly what I meant, but "let" is always appropriate and you can begin a proof by contradiction with it. Using "suppose" just gives the reader an additional cue about what you intend to do with the assumption. $\endgroup$
    – Karl
    Mar 8, 2021 at 1:39
  • $\begingroup$ One last question. When you make the assumption that x is an integer, do you think of this as "Let's just pretend that the symbol x (which has no representation beforehand) happens to represent an integer"? – Pranav Jain 8 mins ago Delete $\endgroup$ Mar 8, 2021 at 1:43
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    $\begingroup$ I agree with the other commenters that "pretending" seems unnecessary, but there's no harm in thinking of it that way. You're introducing an assumption $x\in\Bbb Z$ that happens to feature a new symbol $x$. From this assumption you might draw more conclusions involving $x$, and you might then generalize your conclusions to get a "for all" statement that says something about every integer. $\endgroup$
    – Karl
    Mar 8, 2021 at 1:49
  • $\begingroup$ The statement "You're introducing an assumption x∈ Z that happens to feature a new symbol x, and from this you might draw more conclusions involving x." sums up exactly my thought process. Also, I was interested in applying this to "for all statements". Thank you for all your help! $\endgroup$ Mar 8, 2021 at 1:54
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You should take "Let $x$ be an integer" as a synonym for "Assume that $x$ is an integer" or "Consider an integer $x$".

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  • $\begingroup$ When you say "Assume that x is an integer", do you think of this as ""Let's just pretend that the symbol x (which has no representation beforehand) happens to represent an integer"? $\endgroup$ Mar 8, 2021 at 1:32
  • $\begingroup$ @PranavJain Yes. If you use a symbol like $x$, you must explain what it represents in what follows. Another way to say it would be "we use the symbol $x$ to denote an integer". $\endgroup$
    – Paul Frost
    Mar 8, 2021 at 9:56

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