Problems 01
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Problem Solving Set 1 04 July 2012 1. Let f ∈ C 1 (a, b), lim x→a+ f (x)=+∞, lim x→b- f (x)= -∞ and f 0 (x)+ f 2 (x) ≥-1 for x ∈ (a, b). Prove that b - a ≥ π and give an example where b - a = π. 2. A collection of subsets of {1, 2,...,n} has the property that each pair of subsets has at least one element in common. Prove that there are at most 2 n-1 subsets in the collection.
description
Examination sample, of the mentioned university
Transcript of Problems 01
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Problem SolvingSet 1
04 July 2012
1. Let f C1(a, b), limxa+ f(x) = +, limxb f(x) = and f (x)+ f 2(x) 1 for x (a, b). Prove that ba and give an example where b a = .
2. A collection of subsets of {1, 2, . . . , n} has the property thateach pair of subsets has at least one element in common.Prove that there are at most 2n1 subsets in the collection.
