Answer:
See proof below
Explanation:
(A \ B) also represented by (A - B) is the difference of sets A and B in that specific order.
(A \B) is the set of all elements in set A which are not in set B. A Venn diagram can represent this pictorially and make it easier to understand
See the attached image
As you can see the shaded portion represents (A \ B)
One of the properties of the difference of sets is that
(A \ B) = (A ∩ B') where B' is the complement of B i.e. the set of all elements not in set B. This is easily seen as true in the Venn diagram
(A \ B) U (A U B) = (A ∩ B') U (A U B)
= (A ∩ A) u (B' ∩ B) by the distributive law
But B ∩ B' is the null set Ф since a set and its complement are disjoint
So B' ∩ B = Ф
A ∩ A = A ==> the intersection of a set with itself is that set
So we get
(A \ B) U (A U B) = (A ∩ A) ∪ (B' ∩ B = (A U Ф)
The union of any set with the null set is just the set
(A \ B) U (A U B) = A U Ф = A
Proved
I hope that explains it. Please ask for any clarifications if anything is confusing in my explanation