Question

If a boy throws up a ball with initial velocity, and if the (lets say) elevator goesup by an acceleration 'a', will the net acceleration on that ball be a-g? Please explain.

Answer 1

No, it won't. Because when the elevator moves, there are three types of forces that are applied on it. 
1. gravitational force (mg)
2. The force by which it's moving upwards (ma)
3. illusory force (ma₁) (in the opposite direction of ma) 
so, [using the FBD of the lift] net acceleration of the lift, a₁=a-g

But when the ball moves upwards, there are only 2 types of forces hat are applied on it.
1.Gravitational force (mg)
2.illusory force (ma₁)
And that's why net acceleration, a₁=-g

Hope it helped.