Sets a and b, shown in the venn diagram, are such that the total number of elements in set a is twice the total number of elements in set b. altogether, there are 3011 elements in the union of a and b, and their intersection has 1000 elements. what is the total number of elements in set a?

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:



We are also given that there are 1000 elements in the two sets' intersection. Hence, we get:
and 
The total number of elements combined in set A and set B can be represented as: 

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:

We can simplify this equation to . In the beginning, we found that a = 2b, or b = 1/2a, so we can substitute that into the equation. Here is the process:




Therefore, the total number of elements in set a is 2674.