About two measurable set question

Question

Question:

If $A$ and $B$ are measurable set,

$ m(A) + m(B) = m(A\cup B) + m(A\cap B)$

equality is provided.(Lebesgue Measure)

My Solution:

$m(B) = m(B\cap A) + m(B \cap A^\mathbb{c}) = m(B\cap A) + m(B \setminus A)$

$m(A \cup B ) = m((A \cup B) \cap A) + m((A \cup B) \cap A^\mathbb{c}) = m(A) + m(B \setminus A)$

So, Would it be equal?

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$m((A \cup B) \cap A^\mathbb{c}) = m(B \setminus A)$

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If it's equal

$ m(B \setminus A) = m(A \cup B ) - m(A)$

$ m(B \setminus A) = m(B ) - m(A \cap B)$

hence,

$ m(A) + m(B) = m(A\cup B) + m(A\cap B)$

Answer 1

$$(A \cup B) \cap A^C=(A\cap A^C)\cup (B \cap A^C)=\emptyset\cup (B\setminus A)=B\setminus A$$