jmc

Let $a$ be the number of elements in set $A$ and $b$ be the total number of elements in set $B$. We are told that the total number of elements in set $A$ is twice the total number of elements in set $B$ so we can write $$a=2b.$$ Since there are 1000 elements in the intersection of set $A$ and set $B$, there are $a-1000$ elements that are only in set $A$ and $b-1000$ elements only in set $B$. The total number of elements in the union of set $A$ and set $B$ is equal to $$\text{mbox}elementsinonlyA+\text{mbox}elementsinonlyB+\text{mbox}elementsinboth$$ which we can also write as $$(a-1000)+(b-1000)+1000.$$ Because we know that there is a total of 3011 elements in the union of $A$ and $B$, we can write $$(a-1000)+(b-1000)+1000=3011$$ which simplifies to $$a+b=4011.$$ Because $a=2b$ or $b=\frac{1}{2}a$, we can write the equation in terms of $a$ and then solve for $a$. We get $$\begin{array}{rl}a+b& =4011⟹\\ a+\frac{1}{2}a& =4011⟹\\ \frac{3}{2}a& =4011⟹\\ a& =2674\end{array}$$ Therefore, the total number of elements in set $A$ is $\boxed{2674}.$