What is $(A\setminus B)^c$? It seems to me that it is equal to $B\cup A^c$, isn't it? Please help me to verify if there is a mistake?
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Using the property $A \backslash B = A \cap B^c$ and the De-Morgan's laws you get the following chain of equalities:
$$ (A \backslash B)^c = (A \cap B^c)^c = A^c \cup (B^c)^c = A^c \cup B $$