What if union of disjoint sets results in universal set?

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What if union of disjoint sets results in universal set?

I have a question related to set theory. If $A_1,A_2,A_3\dots, A_n$ belongs to universal set $U$, and if all of the sets are disjoint i.e. $A_i \cap A_j = \emptyset$ for all $i$ and $j$. And If their union equals to Universal set i.e. $A_1 \cup A_2 \cup \dots A_n = U $ = Universal set. What is such situation called?

Answer:

If every $A_i$ is nonempty, you have described a partition of the set $U$. The definition holds for...

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Can a specialized version of the disjoint-set data structure support union and find on a set of intervals in constant time?

Specifically, is there a way to maintain disjoint intervals of integers from 1 to N, while being able to support the following operations in constant time? Union: Merge two adjacent intervals. Find: Determine which interval a specified integer is in.

Answer:

This 1989 paper notes that there exists an amortized constant-time algorithm for the interval union...

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The interior Ǻ of a set A is the union of all open sets included in A?? Maths help?

The interior Ǻ of a set A is the union of all open sets included in A a) prove that, for any set A, its interior Ǻ is open. b) prove that, for any set A, we have Ǻ ⊆ L(A) Please explain :)

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a) To prove this, we need to show that for any a in int(A), there is an open set S such that a is in...

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Answer:

As A1, A2, etc., are countable, there exist injective functions f1:A1→N, f2:A2→N, etc. Then...

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Union just means, write down the set containing all the elements from either of the sets. Intersection...

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