Question

True or false?

a) If ?? ⊆ {1,2,3,…,2019} and |??| = 1010, then ?? contains two different elements whose sum is 2020.

b) If ?? ⊆ {1,2,3,…,2019} and |??| = 1011, then ?? contains two different elements whose sum is 2020.

c) If ?? ⊆ {1,2,3,…,9} and |??| = 7, then ?? contains three different elements whose sum is 15.

d) If ?? ⊆ {1,2,3,…,9} and |??| = 6, then ?? contains three different elements whose sum is 15.

e) Any set consisting of exactly 9 different integer numbers has a non-empty subset, the sum of whose elements is a multiple of 10.

f) Any set consisting of exactly 100 different integer numbers has a non-empty subset, the sum of whose

elements is a multiple of 100.

g) Any non-empty set consisting of exactly ?? different integer numbers has a non-empty subset, the sum of

whose elements is a multiple of ??.

h) Any set consisting of exactly 10 different numbers chosen from the first 106 positive integer numbers

contains two non-empty disjoint subsets, the sum of whose elements is the same.

i) Any set consisting of exactly 7 different numbers chosen from the first 65 positive integer numbers contains

two non-empty disjoint subsets, the sum of whose elements is the same.

Answer 1

Here given that sset 1,2,3, ----,20197 a). let ssc 1 1,2,3 - - 20194. Such that Iss1=1010. aasta take ss = {1, 2, 3, ---, 1010! > Ssc {1, 2, 3, . . - - , 2019h. But there does not exist any two distinct element of Ss. whoes sum is 2020 so statement (a) is false. b. et ssc {1, 2, 3, - - - , 20184 sit Iss1 = 1011 In this case ss alway contains two element 100g & 1011 set 100gf 1011 = 2020 where so = 22, 2, 3, ....., 20114 or ss = {2009, 2010, - - , 20184. so option 6 is true. (OR) in general en such case sconfein two element & & 2020-8 en ss sit rt(2020-2) = 2020

c). let sse/1,2,3, .. – 94 dit 158107 here we define SS={ 1, 2, 3, 6, 7, 8, 940 $1,2,3,4,5,6,7,8,9 But there is no three distinct element in ss sit there sum is equal to 15. So e Statement ③ is false, is silf322 d). let s=4 1,2, 6, 7, 8, 94 asos sere ssc {1, 2, 3 ... 94 2 Lot 1221 &1851-6. 3.1 2 But there is no three distinet element such that their sum is L OOD I. Hence option hot @ is false. S 22 viol 2001:22