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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.

ryang
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Pranav Jain
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  • Comments are not for extended discussion; this conversation has been moved to chat. – Xander Henderson Mar 10 '21 at 03:41
  • 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. – littleO Sep 21 '21 at 19:25
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    Unfortunately, none of the 3 existing answers by Paul Frost, ryang or Karl are really correct. Read this post to grasp the issue fully. – user21820 Oct 23 '21 at 09:52

3 Answers3

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  1. The verb ‘let’ assigns 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.

ryang
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  • 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? – Pranav Jain Mar 08 '21 at 02:20
  • 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. – Pranav Jain Mar 08 '21 at 02:24
  • 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". – Pranav Jain Mar 08 '21 at 02:27
  • 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. – Pranav Jain Mar 08 '21 at 02:36
  • The reason I bring up this "unknown integer" business is because it forces you to make statements that apply only to all integers (because you don't know which integer is assigned to x, you only know that one of the integers has been assigned). – Pranav Jain Mar 08 '21 at 02:42
  • @PranavJain “it forces you to make statements that apply only to all integers” That's exactly the point. Whenever you wish to prove a statement of the form “For all integers, $A\implies B$”, it is quite safe to begin by writing “Let $x\in\mathbb{Z},$ and suppose $A$.” – ryang Mar 08 '21 at 02:47
  • So just to clarify, you wouldn't say "suppose x ∈ Z and suppose A"? – Pranav Jain Mar 08 '21 at 02:50
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    Thank you for your help Ryan. I will make a summary in my answer on what I learned from everybody's reponses. – Pranav Jain Mar 08 '21 at 03:08
  • No worries! Thank you for your help! And feel free to provide any clarifications to my summary. – Pranav Jain Mar 08 '21 at 03:22
  • Let us continue this discussion in chat. – Pranav Jain Mar 08 '21 at 04:45
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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.

Karl
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  • 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"? – Pranav Jain Mar 08 '21 at 01:26
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    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. – Karl Mar 08 '21 at 01:39
  • 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 – Pranav Jain Mar 08 '21 at 01:43
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    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. – Karl Mar 08 '21 at 01:49
  • 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! – Pranav Jain Mar 08 '21 at 01:54
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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 $x$".

Paul Frost
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  • 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"? – Pranav Jain Mar 08 '21 at 01:32
  • @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". – Paul Frost Mar 08 '21 at 09:56