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Question

What is the difference between (ii) and (iv).  I understood (ii) as "All the engineers like at least some doctor"

What will be the translation for (iv) ?

Comments

You might find this useful:

https://math.stackexchange.com/questions/304217/is-forall-x-exists-y-qx-y-the-same-as-exists-y-forall-x-qx-y

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translation for (iv)... there exist at least one doctor who is liked by all the engineers...

see the difference... in (ii), doctors liked by the engineers may be same or different doesn't matter... but there should be at least one for every engineer. whereas in (iv), same doctor should be liked by all the engineers.

Answer 1

2 says: For all x, there exists a y i.e there is one common person for all the people.

4 says:  There exists y for all x i.e for every x there exists one y. The person doesn't have to be common.

It's a subtle difference.