Here $$S = \{ 4,6,9\} $$

And $$T = \{ 9,10,11,\,\,......,\,\,1000\} $$.

We have to find all numbers in the form of $$4x + 6y + 9z$$, where $$x,y,z \in \{ 0,1,2,\,......\} $$.

If a and b are coprime number then the least number from which all the number more than or equal to it can be express as $$ax + by$$ where $$x,y \in \{ 0,1,2,\,......\} $$ is $$(a - 1)\,.\,(b - 1)$$.

Then for $$6y + 9z = 3(2y + 3z)$$

All the number from $$(2 - 1)\,.\,(3 - 1) = 2$$ and above can be express as $$2x + 3z$$ (say t).

Now $$4x + 6y + 9z = 4x + 3(t + 2)$$

$$ = 4x + 3t + 6$$

again by same rule $$4x + 3t$$, all the number from $$(4 - 1)\,(3 - 1) = 6$$ and above can be express from $$4x + 3t$$.

Then $$4x + 6y + 9z$$ express all the numbers from 12 and above.

again 9 and 10 can be express in form $$4x + 6y + 9z$$.

Then set $$A = \{ 9,10,12,13,\,....,\,1000\} .$$

Then $$T - A = \{ 11\} $$

Only one element 11 is there.

Sum of elements of $$T - A = 11$$