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Consider F be a family of all subsets of set {1,2,3,.....100} that contain atleast 50 numbers,partially ordered with respect to containment.Then maximum size of chains in the poset (F,⊆) that cover F is

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F is a set of all subsets of set {1,2,3,..100} with atleast 50 numbers without disturbing the order.

 Now the, smallest such possible subsets will be of length 50 {1,2,..50}, {1,3,4,5,..51} ...till {51,52,..100}.

Set F will contain such subsets of length 50 to 100.

Now, with respect to the $\subseteq$, {1,2,..50} will be subset of {1,2,..51}.

{1,2,..51} will be subset of {1,2,..52}.

Like this we can go till {1,2,..99} will be a subset of {1,2,..100}. The chain from {1,2,..50} to {1,2,..100} will be of size 51. Hence, 51.
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