Hi there,
Let's solve your problem step by step. First off, we need to assign variables to each set. Here's how you do that:
<span>Let a represent set A and let b represent set B.
</span>Now that we have that down, we can move on. Our next step is to translate our given information to numbers. We are given that set A has twice the total number of elements than in set B. This is what we get after the translation:
![a=2b](https://tex.z-dn.net/?f=a%3D2b)
We are also given that there are 1000 elements in the two sets' intersection. Hence, we get:
![a-1000](https://tex.z-dn.net/?f=a-1000)
and
![b-1000](https://tex.z-dn.net/?f=b-1000)
The total number of elements combined in set A and set B can be represented as:
![(a-1000)+(b-1000)+1000](https://tex.z-dn.net/?f=%28a-1000%29%2B%28b-1000%29%2B1000)
The question gives us that there are 3011 total elements in the union of A and B, so we can equate the expression above to 3011. This is our resulting product:
![(a-1000)+(b-1000)+1000 = 3011](https://tex.z-dn.net/?f=%28a-1000%29%2B%28b-1000%29%2B1000%20%3D%203011)
We can simplify this equation to
![a+b=4011](https://tex.z-dn.net/?f=a%2Bb%3D4011)
. In the beginning, we found that a = 2b, or b = 1/2a, so we can substitute that into the equation. Here is the process:
![a+b=4011](https://tex.z-dn.net/?f=a%2Bb%3D4011)
![a+ \frac{1}{2}a=4011](https://tex.z-dn.net/?f=a%2B%20%5Cfrac%7B1%7D%7B2%7Da%3D4011)
![\frac{3}{2} a=4011](https://tex.z-dn.net/?f=%20%5Cfrac%7B3%7D%7B2%7D%20a%3D4011)
![a=2674](https://tex.z-dn.net/?f=a%3D2674)
Therefore, the total number of elements in set a is
2674.