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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

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    $\begingroup$ “Define a relation ... by” in math terminology means “We are telling you that the definition of the relation ... is.” $\endgroup$ – Steve Kass Jul 25 at 22:30
  • $\begingroup$ @steve "Define relation R" sounds like they are asking us to define relation R. $\endgroup$ – Ane Sa Jul 25 at 22:35
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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.

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  • $\begingroup$ Thanks. That "Define relation R" kept confusing me, I thought it is asking to define R (which is already defined, so how do we even define again? Lol). Thanks for your answer. You made it very clear. Thanks. $\endgroup$ – Ane Sa Jul 25 at 22:41
  • $\begingroup$ In a way, the author is telling you to define $R$, but is telling you exactly how to do it, so that you and the author are “on the same page,” so to speak, later. Other mathematical language for this same thing are “Let $R$ be the relation defined by ...” or “Suppose $R = \dots$. I can see how “Define” sounds more like an instruction than the others, but you get used to it. $\endgroup$ – Steve Kass Jul 25 at 23:50
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    $\begingroup$ "Let R be ..." is imperative also. $\endgroup$ – Andreas Blass Jul 25 at 23:58
  • $\begingroup$ @steve Why is the author telling us to define R when they define it afterwards themselves? $\endgroup$ – Ane Sa Jul 26 at 0:17
  • $\begingroup$ It’s just the way mathematics is traditionally written, that’s all. $\endgroup$ – Steve Kass Jul 26 at 0:56
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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.

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  • $\begingroup$ "technically they are asking you to define it, but they are telling you exactly how" then how do you define it again when they have already defined the relation? Like in the main question, R is already defined as R = {(a,b) : b =a+1)}, so how to define R again? $\endgroup$ – Ane Sa Jul 25 at 22:57
  • $\begingroup$ There is no "again". Think of a step in a baking recipe: "Add egg whites by cracking the eggs one by one, separating the yolk from the white and storing it into the batter carefully" They are not telling you to first add the egg whites, and then crack eggs to make more egg whites and then add those too. They are telling you to add whites, then they explain how the whites are to be produced and added. $\endgroup$ – Arthur Jul 25 at 23:11
  • $\begingroup$ It's the same here. "Define a relation $R$ by $R=\{(a,b):b=a+1\}$" is not first telling you to define a relation $R$, then telling you that there is a second relation called $R$ with pairs $(1,2), (2,3)$ and $(3,4)$. It's telling you to define a relation, that the relation is to be called $R$, and then it tells you what exactly that relation is supposed to be (namely the relation with the pairs listed above). $\endgroup$ – Arthur Jul 25 at 23:11
  • $\begingroup$ @AneSa They aren't defining it. They are telling you to define it, and then they are telling you how you should define it / what you should define it as. $\endgroup$ – Arthur Jul 26 at 0:26

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