If events are independent:
\(P\left( A\cap B \right) =P\left( A \right) \times P\left( B \right)\)
*This would actually produce a tangible value not equal to 0 if events are independent.
Now:
\(P\left( A\cap B \right) =P\left( A \right) +P\left( B \right) -P\left( A\cup B \right)\)
Let's combine these two formulas in order to figure out whether your premise is true or false.
\(P\left( A \right) +P\left( B \right) -P\left( A\cup B \right) =P\left( A \right) \times P\left( B \right) \\ \\ \therefore \quad P\left( A\cup B \right) =P\left( A \right) +P\left( B \right) -P\left( A \right) \times P\left( B \right)\)
This means that the premise is false.
