# Difference in terminology between Let and Assume?

I was writing an solution to a problem in a textbook about how to factor a quadratic equation.

I was told that my use of assume was incorrect and it should have used let; however, my teacher couldn't explain the reason why. Could someone explain the usage difference?

at least tell me of a resource that would explain how to use the different mathematical terminology correctly in English?

The fraction $\frac{x-2}{\left(x+7\right)\left(x-3\right)}$ is not defined when $x$=?

Assume $x=$ -7 or -3, then the denomiator is equal to zero, thus not defined.

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Can you include the sentence you were told was wrong? –  Jim Sep 4 '13 at 1:54
You can assume an hypothesis, you can let something be something. «Let $x=0$...» and «Assume $x=0$...» are radically different things. –  Mariano Suárez-Alvarez Sep 4 '13 at 1:59
From a purely mathematical/logical standpoint, I don't think there's a difference. Both just amount to a conditional. The only difference in use seems to be the context surrounding the problem. –  Ataraxia Sep 4 '13 at 2:01
@Ataraxia: There are situations in which they are interchangeable, but there are situations in which they definitely are not. –  Jim Sep 4 '13 at 2:04
@MarianoSuárez-Alvarez: If I want to prove $\forall x > n, P(x)$ I could start with "assume $x$ is an integer greater than $n$" or I could say "let $x$ be an integer greater than $n$". I personally would use "assume" for stylistic reasons but I wouldn't tell anyone that "let" is wrong in that circumstance. –  Jim Sep 4 '13 at 2:08

Let: This is usually used in a definition, when no earlier context is given. For example, "let $x$ be a real number", "let $n$ be an integer". We would then continue with the proof, and use this definition for whatever we had in mind. This is used in practically every mathematical proof.

Assume: This is used in a statement that elaborates on a previous context. For example, we might say "let $x$ be an real number.... Assume that $x$ is positive" or "let $n$ be an integer... assume that $n$ is even."

"Assume" is often used in order to go through a proof by exhaustion (i.e. a proof by cases) or in order to begin a proof by contradiction. You might also see "assume to derive a contradiction". This can be thought of as interchangeable with the word "suppose", which is more often used in the latter case.

In some instances, you might use "assume" to indicate the statements that you accept without proof. For example, a question might state "you may assume the fundamental theorem of algebra", which would mean that you're allowed to use the fundamental theorem of algebra without proving it yourself.

For the problem you've presented, I would disagree with your teacher's assessment that "let" would be the better word to use here. In any case, I think the clearest way to answer the question would have been:

If $x=-7$ or $x=3$, then the denominator is equal to zero, and thus the fraction is not defined.

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Assume is circumstantial, let is not.

$\bullet$ We use assume to investigate the implications of a certain assumption, given a certain context. For instance, say we want to prove that $A$ is true , but $A$ requires that we consider two cases, $a$ and $a'$. Then we can assume $a$ is true, observe that it implies $A$, then assume $a'$ is true, and observe that it implies $A$ as well. Thus we have shown $A$ is true, by assuming all possible cases. Another word we can use is suppose. The idea is that you are considering a possibility, whether it leads to something true or not.

Note that this depends entirely on the problem, hence why I use the word circumstancial: for instance, you cannot assume $f(x)=\ln x$ if neither $f$ nor $\ln x$ appear in the problem.

$\bullet$ We use let when we wish to introduce a tool which will enable us to solve the problem. It is permanent throughout the proof, and is more of a commodity. For instance, if we want to prove $A$ is true, and need to use a certain mathematical object to do so, then it is useful to let: "something" be "said object". Another word for this is $set$.

This is not circumstantial, in the sense that one could $let:$ "something" be "said object" under any circumstance, hypothetically. For example, one can let $f(x)=\ln x$ no matter what the context of the problem is.

Example

We want to hike to the top of a mountain via the quickest route. There are exactly four possible routes. We may let these routes be $a,b,c,d$ so that we may refer to them later in the solution. Next, to solve the problem, we may assume that we take $a$, observe that it would take a certain time, then assume that we take $b$, etc. and conclude.

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"Let" defines a new symbol. This symbol can represent a specific object or an arbitrary one, as in "let $p$ be the smallest prime factor of $x$" or "let $p$ be an arbitrary prime".

"Assume" indicates that you are introducing a hypothesis to an if-then argument. This can be a hypothesised new object or it can be a property you are demanding an existing object hold, as in "assume there is an integer $n$ such that $x < n < y$" or "assume $x$ is divisible by $2$".

As to your specific sentence, I would have said "if $x$ equals $7$ or $-3$ then ...". I wouldn't have used "assume" or "let".

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Why would you have used "if"? I am not trying to have a flamewar, but math is hard enough, but the english is killing me. –  yiyi Sep 4 '13 at 2:10
One way to prtect yourself from English is to read a lot of good English. Can you find an example of the construction «Asume $x=1$, then something» in a respectable English text? –  Mariano Suárez-Alvarez Sep 4 '13 at 2:14
I can't think of a rule to give you for why "if" sounds better to my native ears then "assume" or "let" does, sorry :( –  Jim Sep 4 '13 at 2:16
@yiyi One reason why "if" may sound better to a native speaker is that "if" is typically more definite than "assume." When we write "assume x = 2" in everyday writing, this implies that we don't really know if x=2 or not, but we are just believing it to be so. When we write "if x=2," then we know that "x=2" every time we do the thing referenced in the "then" part of the sentence. HOWEVER: If vs. assume is something that is really splitting hairs. Saying "if" is probably something a native speaker would do naturally, but "assume" would be a perfectly acceptable form, too. –  anorton Sep 4 '13 at 3:03
@ruakh: I suspect the "then" there is temporal, not causal. anorton is not affirming the consequent. –  Jim Sep 4 '13 at 5:16

"Let" is usually used as a precursor to an if/then statement, and an establishment of data that is not axiomatic. "Assume" is almost always implying an axiom or some statement of proof without proof. The distinction here is miniscule, and I do not think it is significant mathematically but in terms of pure style I can see the point of using one word over the other.

To illustrate the subtle difference I'll use an example.

If I say "Let $f$ be a quadratic polynomial then $f$ is a polynomial", I am letting $f$ be arbitrary and not making an assertion about some $f$. Simply I am saying if $f$ is quadratic, then $f$ is also a polynomial.

If I instead say "Assume $f$ is a quadratic polynomial, then $f$ is a polynomial" I am making some assertion about $f$. I am no longer stating "if $f$ is quadratic, then $f$ is also a polynomial", but instead saying the following. I have an $f$. I assume $f$ is quadratic (I take it as truth that $f$ is quadratic), then $f$ is a polynomial.

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sorry, English is not my primary language. Could you add more to the last two paragraphs? –  yiyi Sep 4 '13 at 2:06

I don't know of any resource that discusses this convention, but I can describe how I think it works.

One uses "assume" to prove implications. For example:

Theorem. If A, then B.

Proof. Assume A holds...therefore B holds.

One uses "let" to prove universal statements. For example:

Theorem. For every $\epsilon > 0$ there is a $\delta > 0$ such that...

Proof. Let $\epsilon > 0$...

EDIT: In your particular case you might write the following (probably less formally):

Theorem. If $x = -7$ or $x = 3$, then...

Proof. Assume that $x = -7$ or $x = 3$...

So I would use either "if" or "assume" in this case. I don't know why your teacher wanted "let."

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To "let" is to define for the purposes of the exercise. For instance, "Let $x = -7$". Within the context of this exercise, $x$ now unequivocally equals negative seven, and you can never say or show differently because that reduces to absurdity; if you ever showed within the same mathematical exercise that $x=6$, then $-7=6$ which is nonsensical. "Let", therefore, is to be taken as gospel while reading the proof, and if it's ever seen to be false, that is a contradiction invalidating the proof (in many cases proving the opposite).

To "assume" is to reason based on a conditional proposition. The assumption may or may not actually be true; we are "assuming" that the statement is a true statement, and if it is true, then something else is also true. The original assumption may in fact be false in a specific situation. That simply means that the lemma based on the assumption is no longer valid in this situation; it is not in itself a disproof of the lemma as it pertains to the assumed situation, unless it can be independently proven that the assumption can never be true (so, for instance, any proof based on "Assume 1=2" is nonconstructive).