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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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but why not 100?
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yes it should be 51

+1 vote
An antichain  in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable.

A maximum or longest chain is one which is of the greatest size possible. The size of the longest chain is known as a poset’s height.The following gives an example of such a chain

{1, 2, . . . , 50} ⊂ {1, 2, . . . , 51} ⊂ · · · ⊂ {1, 2, . . . , 100}

which is the maximum size of chain i.e. 51.Hence, the answer is 51.
by Active (1.7k points)
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comparable mean?
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x and y are comparable if x ≤ y and/or y ≤ x hold
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ok 50 to 100- total 51 subsets