SOLUTION: it is question from set theory. I can't write the notations clearly because they are not programmed on my phone . so please bear me . the question goes follows: (A-B)UB=AUB.

Question 1013595: it is question from set theory. I can't write the notations clearly because they are not programmed on my phone . so please bear me . the question goes follows: (A-B)UB=AUB.
Answer by ikleyn(52778) (Show Source): You can put this solution on YOUR website!
.
It is question from set theory. I can't write the notations clearly because they are not programmed on my phone. So please bear me. the question goes follows: (A-B)UB=AUB.
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First, explanation.
The set A-B is the set of those elements of the set A that do not belong to the set B.

  It may happen that you just now what is A-B. 
  So, this explanation is just in case.

Now let us prove that each element of the set (A-B)UB belongs to AUB.

Actually, it is absolutely clear, because the set (A-B)UB is simply a subset of AUB.


Next, let us prove that each element of the set AUB belongs to (A-B)UB.

Let x belongs to AUB. 
Then x belongs to A or x belongs to B or x belongs to both A and B.
In any case x belongs to (A-B) or B.

Thus we proved that each element of the (A-B)UB belongs to AUB and , in opposite, each element of the AUB belongs to (A-B)UB. 
Hence, x belongs to (A-B)UB.

Thus these sets consist of the same elements.
Therefore, these sets are identical.

Is everything clear to you?

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