How to find all the pairs satisfying a relation?

Question

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.

Answer 1

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?

Answer 2

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

Answer 3

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.