When do we use "suppose" and when "let"?

As a matter of fact, these two words are used a lot in mathematical contexts. Often, we use them interchangeably; but I do realize that that might not be correct. What should I do about this matter? To be honest, a student of mine has asked me about these two words and when to use each. Any help?

The difference in connotation is that we usually use suppose to make an assumption and let to make a declaration.

Suppose X is true. It follows that Y is false.

We let Z be an integer so that 2Z+1 is odd.

In mathematical writing, you may often replace suppose with assume or, more loosely, pretend that you already know or pretend that it is true. You can use it to specify a condition or allow for a hypothetical situation. For example, you may see something like,

In order to prove that there are infinitely many primes, you can first suppose that there is a largest prime, then derive a contradiction.

or

Suppose that the Riemann hypothesis is true. Then Miller's primality test is deterministic.

On the other hand, to let usually means to set the definition. For example,

Let N be the set of natural numbers and Q be the set of rational numbers. Then N has the same cardinality as Q.

Often, you can use suppose in place of let, because you want the reader to assume a certain definition or notation. The converse does not always hold, that is, you cannot always use let in place of suppose.

Rule of thumb: Suppose is used for assuming the truth value of a statement or proposition. Let is used for assigning a mathematical value to a symbol.

"Suppose N is finite" has meaning while "Let N be finite" doesn't make sense. On the other hand, both "Let n = 1" and "Suppose n = 1" are acceptable, though the former is preferred. I can see a situation when you might use the latter.