I am stuck with a question about relations which I have stated below.

$A$ and $B$ are sets of real numbers and $aRb$ iff $2a+3b=6$. Find the domain and range of $R$.

Now the problem I am facing is that there can be numerous pairs which satisfy this relationship. Here are some,

$(0,2), (3,0), (2,3/2), \ldots$

But the problem is that how will I calculate the domain and range for this? Is there any other way I can find the domain and range? Or maybe another way to find all the pairs that satisfy the relation?

Thanks in advance.

  • 1
    $\begingroup$ I have made some edits, primarily setting the math in latex and also removing the scroll bar that was present. I also edited the question, since I thought "relation" is a crisper word than "relationship" which doesn't have a formal meaning. Hope these edits are ok. They will be visible only after they are peer-reviewed. $\endgroup$ – Srivatsan Jul 31 '11 at 4:42
  • $\begingroup$ The question isn't well defined as it stands. Everyone has answered this question as if R is a relation defined on the reals, but it seems to me as if R is meant to be defined on A and B. Now, if A = B = [0,1], for example, the situation is very different! $\endgroup$ – Billy Jul 31 '11 at 5:29

I would like to ask the following questions to "answer" your question:

  • What's the definition of relations? And what's $R$ in your case?
  • What are the definitions of "domain" and "range" for a relation? Can you see how the definitions work in your question?
  • $\begingroup$ Relations are sets having some link within. Here all the points satisfying the equation belong to the relation. Domain being all the 'a' values and Range is all the 'b' values of the relation.Oh, I got it :) Both of them should be equal to the set of real numbers ? $\endgroup$ – user2857 Jul 31 '11 at 5:12
  • $\begingroup$ @fahad: According to this wiki article about binary relation, yes. $\endgroup$ – Jack Jul 31 '11 at 15:36

Hint: Can there be an $x$ such that $\forall y \in \mathbb {R}: (x,y) \not \in R$?

  • $\begingroup$ No, there can not be a 'x' whose pair does not satisfy this relation. But still the problem remains about how do I find its domain and range? $\endgroup$ – user2857 Jul 31 '11 at 4:51
  • $\begingroup$ As it has infinite values so shouldn't the domain and range be infinity? $\endgroup$ – user2857 Jul 31 '11 at 4:52
  • $\begingroup$ @fahad Domain and Range of a relation are sets, not numbers (I am thinking of $\infty$ here as a number). You are thinking of the cardinalities of the domain and range, not the sets themselves. $\endgroup$ – Srivatsan Jul 31 '11 at 4:55
  • $\begingroup$ @fahad Your question should probably read "As it has infinitely many values, shouldn't the domain and range be infinite?", the important corrections being the changes of the use "infinite" and "infinity". In the first, it doesn't have "infinite values", which would mean values that are infinitely large (such elements don't exist in this context). In the second, the domain and ranges are not "infinity", which is a noun describing some mathematical object, but "infinite", which is an adjective describing how many (cardinality as Srivatsan said). So yes, the domain and range are infinite, but... $\endgroup$ – Quinn Culver Jul 31 '11 at 5:08
  • $\begingroup$ @fahad ...that still doesn't tell you what they are, which is what your question asked. (Note that when I said "Your question should probably read...", I was referring to your question here in the comments below Ross's answer, not your original question.) $\endgroup$ – Quinn Culver Jul 31 '11 at 5:09

For any a you give me, I can give you a b value such that 3b + 2a = 6. In fact, if you give me $a$, then I can give you $\dfrac{ 6 - 2a}{3}$ as b. And it will satisfy your relationship. So both a and b are unbounded.

These are the exact values that fall on the line $y = 2 - \frac{2}{3} x$. In fact, this is even a motivating definition of what it means to be a line.

  • $\begingroup$ Thankyou for the informative answer. $\endgroup$ – user2857 Jul 31 '11 at 14:33

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