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Source of this question and problem:

https://www.javatpoint.com/automata-conversion-from-nfa-with-null-to-dfa

This is the question:

enter image description here

enter image description here

So in this figure there are my 1 confusion-:

  1. Don't we need a empty set like this-:

enter image description here

Source: https://www.youtube.com/watch?v=dmvJp5EgOek&list=PLENQMW_c1dimRCKF3bjUqHaH8dvJkapSw&index=48

Since this is DFA we need to show all transitions, isn't it?

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You are correct! As you say, in a DFA you need to show all transitions. So, you have to include the empty set state (sometimes called a "sink" state or "dead" state).

Actually this is just the short answer -- the long answer is that some authors have defined DFAs so that they don't need to include all transitions, but that definition is not common anymore in modern textbooks. So, the source of the first DFA you showed is probably using that (older) definition.

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