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CHAPTER I [:] Notions from Set Theory [...] §2. OPERATIONS ON SETS. "end quote
These pages assume that the reader has the book and is reading along.
topic index     contents
Please read these notes from the beginning to here, and then paragraph 6 of §2.

THE INTERSECTION OF THE COMPLEMENT = THE COMPLEMENT OF THE UNION

Please note that, although the exercise has two sub paragraghs, I count the entire exercise as part of the 6th paragraph.

The exercise is to prove that the intersection of complements is equal to the complement of the union. To apply the definition of the equality of sets to the present proof, it is convenient to express it as follows - where the first double arrow has been made vertical for easier reading:

Rosenlicht proves the bottom line, and then the up direction of the vertical double arrow establishes the desired result:

In the first sub paragraph, we suppose that   x   is in the intersection of complements, which means that   x   is in both complements, and therefore   x   is not in either of the sets   X   or   Y. Then we know that   x   is not in the union of the two sets   X   or   Y. Therefore, we have the required result that   x   is in the complement of the union.

In the second sub paragraph, we suppose that   x   is in the complement of the union. Then   x   is not in the union of the two sets   X   or   Y,   so that   x   is not in either   X   or   Y. This means that   x   is in both the complement of   X   and the complement of   Y. Therefore, we have the required result that   x   is in the intersection of complements.