Suppose I am given a set of S partition of universal set A, define pair as (a,b) iff elements a,b are members of one of set in at least one partition. I want to know how many pairs in S.
Example:
$$A = \{1,2,3,4,5\}$$
$$S = \{
s_1=\{\{1,2\},\{3,4,5\}\},
s_2=\{\{1\},\{2,3,4,5\}\},
s_3=\{\{1\},\{2\},\{3\},\{4,5\}\}
\}$$
Pairs:
(1,2)
(2,3)
(2,4)
(2,5)
(3,4)
(3,5)
(4,5)
Number of pairs = 7
Is there an algorithm that takes less than $O(n^2)$,
$n=|A|$ or correct keywords or name of this problem?
2026-03-26 14:22:13.1774534933
number of pairs that are member of a class of one of the given partitions
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