'How can the union of X and Y be equal to the intersection of Y and Z (X u Y = Y n Z)?
On which conditions can the equation X u Y = Y n Z be true?
I need to prove this equation step by step, but I do not know where to start.
Solution 1:[1]
X union Y = Y intersect Z if and only if X is a (possibly improper) subset of Y and Y is a (possibly improper) subset of Z.
To prove this, we need to show the implication in both directions.
If X is a (possibly improper) subset of Y and Y is a (possibly improper) subset of Z, is it true that X union Y = Y intersect Z? Because X is a subset of Y, X union Y = Y. Because Y is a subset of Z, Y intersect Z = Y. Thus, X union Y = Y = Y intersect Z, as required.
If X union Y = Y intersect Z, is it true that both X is a subset of Y and Y is a subset of Z? Suppose X were not a subset of Y. That means there is some x in X not in Y. So, X union Y must contain x. But Y intersect Z cannot contain anything not in Y, such as x; therefore, X union Y cannot equal Y intersect Z - a contradiction. Suppose instead Y were not a subset of Z. Then Y would contain some element y not in Z. But then Y intersect Z does not contain y, whereas X union Y must. So, X union Y cannot equal Y intersect Z, a contradiction.
This completes the proof that X union Y = Y intersect Z iff X is a (possibly improper) subset of Y and Y is a (possibly improper) subset of Z.
Note: X is a proper subset of Y if X is a subset of Y but X is not equal to Y. If X equals Y, it is an "improper" subset.
Sources
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Source: Stack Overflow
| Solution | Source |
|---|---|
| Solution 1 | Patrick87 |