Let A_1, A_2, ldots, A_m be non-empty subsets | vedclass

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(A) Let $|A_i| = n_i$ for $i = 1, 2, \ldots, m$. Since the sets are pairwise disjoint and are subsets of a set with $100$ elements,the sum of their si

Vedclass · Accepted Answer

$13$

Vedclass · Answer

(A) Let $|A_i| = n_i$ for $i = 1, 2, \ldots, m$. Since the sets are pairwise disjoint and are subsets of a set with $100$ elements,the sum of their sizes must satisfy:$\sum_{i=1}^{m} |A_i| \leq 100$.Since $|A_i|$ are distinct positive integers,to maximize $m$,we choose the smallest possible distinct positive integers for $|A_i|$.Thus,we require $1 + 2 + 3 + \ldots + m \leq 100$.The sum of the first $m$ natural numbers is $\frac{m(m+1)}{2}$.So,$\frac{m(m+1)}{2} \leq 100$,which implies $m(m+1) \leq 200$.For $m = 13$,$13 	imes 14 = 182 \leq 200$ (True).For $m = 14$,$14 	imes 15 = 210 > 200$ (False).Thus,the maximum value of $m$ is $13$.

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