Originally Posted by
matthayzon89 Prove or disprove: For all sets A and B, (A-B) U (A intersect B) = A
My attempt at a proof:
This statement is true.
Suppose that A and B are sets where (A-B) U (A intersect B) and x is an element (A-B) U (A intersect B).
By definition of Union x is an element A-B OR x is an element of A intersect B (but not both).
So if x is an element of (A-B) then by definition of difference x is an element of A and x is NOT an element of B. Therefore, A-B=A.
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If x is an element of A intersection B then x is an element of A and x is an element of B by definition of intersection.
Therefore, x is an element of A intersect B.
So, if x is an element of A intersect B then x is NOT and element of A-B and if x is an element of A-B then x is NOT an element of A intersect B.
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A-B=A because by definition of difference if x is an element of A-B then x is an element of A but it is NOT an element of B. So, x is an element of A which equals A.
If that is NOT the case, then,
x is must be an element of A intersect B, which means it is an element of A and an element of B, so either way if x is an element of B or not an element of B it is always an element of A and a set is always a subset of itself. Therefore, the original statement IS true and for all sets A and B, (A-B) U (A intersect B) = A
END OF PROOF.