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Last updated at May 29, 2018 by Teachoo

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Example 13 Check whether the following statement is true or not. If x, y ∈ Z are such that x and y are odd, then xy is odd. Statement : If x & y ∈ Z are such that x & y are odd, then xy is odd Let p : x , y ∈ Z such . that x and y are odd q : xy is odd That given statement is of the form if p ⇒ q We check the validity of the given statement Statement : If x & y ∈ Z are such that x & y are odd, then xy is odd Method 1 :- Direct Method By assuming that p is true, prove that q must be true. i.e. Assuming x & y are odd , prove that xy is odd Since x & y are odd Let x = 2m + 1 When m, n ∈ Z & y = 2n + 1 Calculating xy = (2m + 1 ) (2n + 1) = 2m (2n + 1 ) + 1 (2n + 1 ) = (2m) (2n) + 2m + 2n + 1 = 4mn + 2m + 2n + 1 = 2 (mn + m + n) + 1 This shows xy is odd Hence q is true. Therefore the given statement is true Method 2 :- Contrapositive Method Statement : If x & y ∈ Z are such that x & y are odd, then xy is odd By assuming that q is false, prove that p must be false. Let xy be not odd , prove that x and y are not odd Since xy is not odd ⇒ xy is even. ⇒ This is possible only if either x or y is even Let us take an example

Proving True - If and only if

Chapter 14 Class 11 Mathematical Reasoning

Concept wise

- Statements
- Writing negation of statements
- Negation - Checking if true or not
- Finding Compound Statement
- Words 'And' & 'Or'
- Inclusive and exclusive or
- Quantifiers
- Contrapositive and converse
- If then
- If and only if
- Proving not true/false (by giving counter examples)
- Proving True - If and only if
- Proving True - By Contrapositive
- Proving True - By Contradiction
- Necessary and sufficient condition

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.