# Thread: Proving two sets equal

1. ## Proving two sets equal

Here's the problem:

Let L={(x,y) | 2x+3y=6} and L'={a(3,0) + b(0,2) | a+b=1}
Prove that L=L'

What I've done:
I've picked a few examples, so I believe that this is true. I know that I have to prove that L<L' and L'<L to conclude that L=L'.
I just can't seem to get how to go about doing it.

I think i did the second part:
if (m,n) is in L', then (m,n)=a(3,0)+b(0,2) where a+b=1
so, m=3a and n=2b
but a+b=1, so b=1-a
so (m,n)=(3a,2b)=(3a,2(1-a))
so then I tried to prove that this point is in L:
2x+3y=2(3a)+3(2(1-a))=6a+6-6a=6
I think this proves the second half, L'=L. Does it?

For the first part, I have:
(p,q) is in L, so 2p+3q=6
and I didn't know what to do so I tried to work backwards from the conclusion to get an idea
(p,q)=(p,0)+(0+q)=(p/3)(3,0)+(q/2)(0,2)
does p/3 + q/2 =1? but I can't figure out how to get any further.

Any help or hints or ideas or anything would be greatly appreciated!

2. Well, the first part is to pick a point in L and show that it is in L', and the second part is to pick a point in L' and show that it is in L. That's how I've always learned to prove that one set is equal to another. Does that make sense?

3. Originally Posted by sfitz
Well, the first part is to pick a point in L and show that it is in L', and the second part is to pick a point in L' and show that it is in L. That's how I've always learned to prove that one set is equal to another.
That is the correct method.
$\displaystyle \begin{array}{l} 2p + 3q = 6 \\ \frac{p}{3} + \frac{q}{2} = 1 \\ \frac{p}{3}\left( {3,0} \right) + \frac{q}{2}\left( {0,2} \right) = \left( {p,q} \right) \\ \end{array}$

4. Originally Posted by Plato
That is the correct method.
$\displaystyle \begin{array}{l} 2p + 3q = 6 \\ \frac{p}{3} + \frac{q}{2} = 1 \\ \frac{p}{3}\left( {3,0} \right) + \frac{q}{2}\left( {0,2} \right) = \left( {p,q} \right) \\ \end{array}$
I'm not sure how to go from the first line to the second two. What am I missing?

5. Originally Posted by sfitz
I'm not sure how to go from the first line to the second two. What am I missing?
You are missing division by 6.

### prove that two sets are equal

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