# Example 19 - Chapter 14 Class 11 Mathematical Reasoning

Last updated at Feb. 17, 2020 by Teachoo

Last updated at Feb. 17, 2020 by Teachoo

Transcript

Example 19 Using the words necessary and sufficient rewrite the statement The integer n is odd if and only if n2 is odd . Also check whether the statement is true. The necessary and sufficient condition that the integer n be odd is n2 must be odd. Let p and q denote the statements p : the integer n is odd. q : n2 is odd. Now checking whether statement be true Care l :- Direct method If p then q i.e. p q It integer n is odd. Then prove that n2 is odd let n = 2k + 1 k Z squaring both side. n2 = (2k + 1 ) n2 = (2k)2 + (1)2 + 2k + 1 = 4k2 + 1 + 4k = 4k2 + 4k + 1 = 4 ( k2 + 1 ) + 1 n2 is odd Hence p = q Case 2 :- Contraption If n2 is odd to prove n is odd we check this by Contrapositive method let n is not odd prove that n2 is not odd i.e. prove that n2 is even i.e. prove that n2 is even Now let n is not odd i.e. n is even i.e. n = 2k Squaring both side n2 = (2k)2 n2 = 4k2 This show n2 is even Hence n2 is not odd

Examples

Example 1
Important

Example 2

Example 3

Example 4

Example 5 Important

Example 6

Example 7 Important

Example, 8 Important

Example 9

Example 10

Example 11

Example 12

Example 13 Important

Example 14 Important

Example 15

Example 16

Example 17

Example 18 Important

Example 19 Important You are here

Example 20 Important

Chapter 14 Class 11 Mathematical Reasoning

Serial order wise

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.