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Question: question 9 let s be a set consisting of 19...

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Question 9: Let S be a set consisting of 19 two-digit integers. Thus, each element of S belongs to the set 10, 11,...,99) Use the Pigeonhole Principle to prove that this set S contains two distinct elements r and y, such that the sum of the two digits of r is equal to the sum of the two digits of y. Question 10: Let S be a set consisting of 9 people. Every person r in S has an age age(a) which is an integer with 1 age(x) < 60. .Assume that there are two people in S having the same age. Prove that there exist two subsets A and B of S such that (i) both A and B are non-empty, (ii) An B- and (iii) ΣΕΑ age(r) = ΣΕΒ age(z). .Assume that all people in S having different ages. Use the Pigeonhole Principle to prove that there exist two subsets A and B of S such that (i) both A and B are non-empty, and (ii) ΣΖΕΑ age(x)-ΣΧΕΒ age(2) ·Assume that all people in , having different ages. Prove that there exist two subsets A and B of S such that (i) both A and B are non-empty, (ii) AnB-, and (ii) ΣΕΑ age (x)-ΣΕΒ age (z).

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