Two shooters

The problem

Two shooters fire to the target. Event A is the first shooter hit the target. Event B second shooter hit the target.

Write equations for the next events:

Solution

$$\\ C = A \cap B \\ D = A \cap B \cup \bar{A} \cap B \cup A \cap \bar{B} \\ E = A \cap \bar{B} \cup \bar{A} \cap B$$