SOLUTION: please help me solve this : A union B = A intersection B implique A = B

Question 1169634: please help me solve this :
A union B = A intersection B implique A = B
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.

            (1)   First, I will prove that every element "x" which belongs to set A, belongs to set B, too.

            (2)   Next, I will prove that every element "y" which belongs to set B, belongs to set A, too.

                    It will mean that the sets A and B are identical.

    Indeed, let x belongs to A.

    Then it belongs to (A union B).

    Due to condition, it implies that x belongs to (A intersection B).

    In turn, it means that x belongs to B.

    First statement is proved.

Now I prove the second statement (the proof is the same due to symmetry)

    Let y belongs to B.

    Then y belongs to (A union B).

    Due to condition, it implies that y belongs to (A intersection B).

    In turn, it means that y belongs to A.

    Second statement is proved.

With two statements proven, the posed problem is solved.

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