What is meant by "Define a relation"?

Question

In my book

Ques. Let A= {1,2,3}, B= {2,3,4}. Define a relation R from A to B by R= {(a,b) : b= a+1}.
i) Write R in roster form.
ii) Write domain, range, codomain.

Answer is given as-

Ans. i) R= {(1,2), (2,3), (3,4)}
ii) Domain = A, Range= B, Codomain= B

I am confusion by "Define a relation R from A to B..."

Y1. Is this asking us to define relation R? Then how do we define relation R (because it already is defined in the question as R= {(a,b) : b= a+1} ).

Y2. Or is it a way to say " Defined is a relation R from A to B by R= {(a,b) : b= a+1}"?

If Y2 is correct, then why do we use this language "Define a relation" instead of "Defined is a relation R from A to B" or "Relation R is defined from A to B"?

Thanks for your help

Answer 1

When a book or problem says "Define relation R by..." followed by a description of the relation, what is meant is "Consider the relation R which is defined as follows:...".

The question is not asking you to define a relation, it is telling you "This is the relation we are concerned with." So your Y2 is the correct interpretation.

You can quibble a bit with the use of the imperative case ("Define R" rather than saying "Let R be a relation defined by" but you must admit that the second way of saying it is more wordy.

Answer 2

Technically they are asking you to define it, but they are also telling you exactly how (by giving you an explicit description). Mathematical texts can read like recipes: "Define this thing. Put the things you have together in that way. Observe the result."

The idea of phrasing it this way is that when someone has found an argument that allows them to reach some conclusion, they want to share their argument with the rest of the world. So they write up the steps, and you are supposed go through it on your own and verify that you get the same result they do by following their "recipe" (also known as a "proof"). And after that, it's only natural that exercises adopt a very similar language to full proofs.