When do we use “suppose” and when “let”?

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.