Example 19 - Using the words 'necessary and sufficient' rewrite - Necessary and sufficient condition

Example 19 - Chapter 14 Class 11 Mathematical Reasoning - Part 2
Example 19 - Chapter 14 Class 11 Mathematical Reasoning - Part 3

  1. Chapter 14 Class 11 Mathematical Reasoning (Deleted)
  2. Serial order wise
  3. Examples

    Example 1 (i) Deleted for CBSE Board 2022 Exams

    Example 1 (ii) Important Deleted for CBSE Board 2022 Exams

    Example 1 (iii) Deleted for CBSE Board 2022 Exams

    Example 1 (iv) Important Deleted for CBSE Board 2022 Exams

    Example 1 (v) Deleted for CBSE Board 2022 Exams

    Example 1 (vi) Deleted for CBSE Board 2022 Exams

    Example 2 (i) Important Deleted for CBSE Board 2022 Exams

    Example 2 (ii) Deleted for CBSE Board 2022 Exams

    Example 3 (i) Deleted for CBSE Board 2022 Exams

    Example 3 (ii) Important Deleted for CBSE Board 2022 Exams

    Example 3 (iii) Deleted for CBSE Board 2022 Exams

    Example 3 (iv) Deleted for CBSE Board 2022 Exams

    Example 4 (i) Deleted for CBSE Board 2022 Exams

    Example 4 (ii) Deleted for CBSE Board 2022 Exams

    Example 4 (iii) Important Deleted for CBSE Board 2022 Exams

    Example 4 (iv) Important Deleted for CBSE Board 2022 Exams

    Example 5 (i) Deleted for CBSE Board 2022 Exams

    Example 5 (ii) Important Deleted for CBSE Board 2022 Exams

    Example 5 (iii) Deleted for CBSE Board 2022 Exams

    Example 5 (iv) Important Deleted for CBSE Board 2022 Exams

    Example 5 (v) Deleted for CBSE Board 2022 Exams

    Example 5 (vi) Important Deleted for CBSE Board 2022 Exams

    Example 6 (i) Deleted for CBSE Board 2022 Exams

    Example 6 (ii) Important Deleted for CBSE Board 2022 Exams

    Example 6 (iii) Deleted for CBSE Board 2022 Exams

    Example 7 (i) Deleted for CBSE Board 2022 Exams

    Example 7 (ii) Important Deleted for CBSE Board 2022 Exams

    Example 7 (iii) Deleted for CBSE Board 2022 Exams

    Example 7 (iv) Important Deleted for CBSE Board 2022 Exams

    Example 8 (i) Deleted for CBSE Board 2022 Exams

    Example 8 (ii) Important Deleted for CBSE Board 2022 Exams

    Example 8 (iii) Deleted for CBSE Board 2022 Exams

    Example 9 Deleted for CBSE Board 2022 Exams

    Example 10 Important Deleted for CBSE Board 2022 Exams

    Example 11 (i) Deleted for CBSE Board 2022 Exams

    Example 11 (ii) Important Deleted for CBSE Board 2022 Exams

    Example 12 (i) Deleted for CBSE Board 2022 Exams

    Example 12 (ii) Deleted for CBSE Board 2022 Exams

    Example 13 Important Deleted for CBSE Board 2022 Exams

    Example 14 Important Deleted for CBSE Board 2022 Exams

    Example 15 Important Deleted for CBSE Board 2022 Exams

    Example 16 Deleted for CBSE Board 2022 Exams

    Example 17 Deleted for CBSE Board 2022 Exams

    Example 18 (i) Deleted for CBSE Board 2022 Exams

    Example 18 (ii) Important Deleted for CBSE Board 2022 Exams

    Example 18 (iii) Deleted for CBSE Board 2022 Exams

    Example 18 (iv) Deleted for CBSE Board 2022 Exams

    Example 19 Important Deleted for CBSE Board 2022 Exams You are here

    Example 20 Important Deleted for CBSE Board 2022 Exams


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 (i) Deleted for CBSE Board 2022 Exams

Example 1 (ii) Important Deleted for CBSE Board 2022 Exams

Example 1 (iii) Deleted for CBSE Board 2022 Exams

Example 1 (iv) Important Deleted for CBSE Board 2022 Exams

Example 1 (v) Deleted for CBSE Board 2022 Exams

Example 1 (vi) Deleted for CBSE Board 2022 Exams

Example 2 (i) Important Deleted for CBSE Board 2022 Exams

Example 2 (ii) Deleted for CBSE Board 2022 Exams

Example 3 (i) Deleted for CBSE Board 2022 Exams

Example 3 (ii) Important Deleted for CBSE Board 2022 Exams

Example 3 (iii) Deleted for CBSE Board 2022 Exams

Example 3 (iv) Deleted for CBSE Board 2022 Exams

Example 4 (i) Deleted for CBSE Board 2022 Exams

Example 4 (ii) Deleted for CBSE Board 2022 Exams

Example 4 (iii) Important Deleted for CBSE Board 2022 Exams

Example 4 (iv) Important Deleted for CBSE Board 2022 Exams

Example 5 (i) Deleted for CBSE Board 2022 Exams

Example 5 (ii) Important Deleted for CBSE Board 2022 Exams

Example 5 (iii) Deleted for CBSE Board 2022 Exams

Example 5 (iv) Important Deleted for CBSE Board 2022 Exams

Example 5 (v) Deleted for CBSE Board 2022 Exams

Example 5 (vi) Important Deleted for CBSE Board 2022 Exams

Example 6 (i) Deleted for CBSE Board 2022 Exams

Example 6 (ii) Important Deleted for CBSE Board 2022 Exams

Example 6 (iii) Deleted for CBSE Board 2022 Exams

Example 7 (i) Deleted for CBSE Board 2022 Exams

Example 7 (ii) Important Deleted for CBSE Board 2022 Exams

Example 7 (iii) Deleted for CBSE Board 2022 Exams

Example 7 (iv) Important Deleted for CBSE Board 2022 Exams

Example 8 (i) Deleted for CBSE Board 2022 Exams

Example 8 (ii) Important Deleted for CBSE Board 2022 Exams

Example 8 (iii) Deleted for CBSE Board 2022 Exams

Example 9 Deleted for CBSE Board 2022 Exams

Example 10 Important Deleted for CBSE Board 2022 Exams

Example 11 (i) Deleted for CBSE Board 2022 Exams

Example 11 (ii) Important Deleted for CBSE Board 2022 Exams

Example 12 (i) Deleted for CBSE Board 2022 Exams

Example 12 (ii) Deleted for CBSE Board 2022 Exams

Example 13 Important Deleted for CBSE Board 2022 Exams

Example 14 Important Deleted for CBSE Board 2022 Exams

Example 15 Important Deleted for CBSE Board 2022 Exams

Example 16 Deleted for CBSE Board 2022 Exams

Example 17 Deleted for CBSE Board 2022 Exams

Example 18 (i) Deleted for CBSE Board 2022 Exams

Example 18 (ii) Important Deleted for CBSE Board 2022 Exams

Example 18 (iii) Deleted for CBSE Board 2022 Exams

Example 18 (iv) Deleted for CBSE Board 2022 Exams

Example 19 Important Deleted for CBSE Board 2022 Exams You are here

Example 20 Important Deleted for CBSE Board 2022 Exams

About the Author

Davneet Singh's photo - Teacher, Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.