MathDB

Determine the smallest possible value of

| A_{1} \cup A_{2} \cup A_{3} \cup A_{4} \cup A_{5} |

, where

A_{1}, A_{2}, A_{3}, A_{4}, A_{5}

sets simultaneously satisfying the following conditions:

(i)

| A_{i}\cap A_{j} | = 1

for all

1\leq i < j\leq 5

, i.e. any two distinct sets contain exactly one element in common;

(ii)

A_{i}\cap A_{j} \cap A_{k}\cap A_{l} =\varnothing

for all

1\leq i<j<k<l\leq 5

, i.e. any four different sets contain no common element. Where

| S |

means the number of elements of

S

Problem Statement