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

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Question 1169634: please help me solve this :
A union B = A intersection B implique A = B
Answer by ikleyn(52803) About Me  (Show Source):
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            (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.