# How to interpret "let" in mathematics?

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.

• Comments are not for extended discussion; this conversation has been moved to chat. Mar 10, 2021 at 3:41
• 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. Sep 21, 2021 at 19:25
• Unfortunately, none of the 3 existing answers by Paul Frost, ryang or Karl are really correct. Read this post to grasp the issue fully. Oct 23, 2021 at 9:52

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.

• 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? Mar 8, 2021 at 2:20
• 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. Mar 8, 2021 at 2:24
• 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". Mar 8, 2021 at 2:27
• 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. Mar 8, 2021 at 2:36
• Thank you for your help Ryan. I will make a summary in my answer on what I learned from everybody's reponses. Mar 8, 2021 at 3:08

"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.

• 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"? Mar 8, 2021 at 1:26
• 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 8, 2021 at 1:39
• 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 Mar 8, 2021 at 1:43
• 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 8, 2021 at 1:49
• 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! Mar 8, 2021 at 1:54

You should take "Let $$x$$ be an integer" as a synonym for "Assume that $$x$$ is an integer" or "Consider an integer $$x$$".

• 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"? Mar 8, 2021 at 1:32
• @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". Mar 8, 2021 at 9:56