Kenneth Rosen Edition 7 Exercise 6.2 Question 14 (Page No. 405)

Question

• A. Show that if seven integers are selected from the first $10$ positive integers, there must be at least two pairs of these integers with the sum $11.$
• B. Is the conclusion in part $(A)$ true if six integers are selected rather than seven?

Answer 1

FOR PART A

We have the 5 sets B=  {1,10}, {2,9}, {3,8}, {4,7}, and {5,6} which has sum of 11, so by Pigeonhole Principle if we choose 7 number it must have at least two set selected.

foe example take one one element from each set A= {1,2,3,4,5} and than select any two number other than set A. after you got two set automatically from set B.

FOR PART B

take six integers  {1, 2,3,4,5, and 6} has just one set {5,6}=11..means only one  set selected but in question  there must be atleast 2 set.