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.
Best Answer
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 a real number $\dots$ Assume that $x$ is positive" or "Let $n$ be an integer $\dots$ Assume that $n$ is even."
"Assume" is often used in order to go through a proof by exhaustion (that is, 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 used more often than "assume" for proofs by contradiction.
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.