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:
Let a represent set A and let b represent set B.
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:
[tex]a=2b[/tex]
We are also given that there are 1000 elements in the two sets' intersection. Hence, we get:
[tex]a-1000[/tex] and [tex]b-1000[/tex]
The total number of elements combined in set A and set B can be represented as:
[tex](a-1000)+(b-1000)+1000[/tex]
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:
[tex](a-1000)+(b-1000)+1000 = 3011[/tex]
We can simplify this equation to [tex]a+b=4011[/tex]. In the beginning, we found that a = 2b, or b = 1/2a, so we can substitute that into the equation. Here is the process:
[tex]a+b=4011[/tex]
[tex]a+ \frac{1}{2}a=4011[/tex]
[tex] \frac{3}{2} a=4011[/tex]
[tex]a=2674[/tex]
Therefore, the total number of elements in set a is 2674.
