Examples

Mathematical Reasoning
Serial order wise

### 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

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.