Example 13 - Chapter 14 Class 11 Mathematical Reasoning (Deleted)
Last updated at May 29, 2018 by Teachoo
Last updated at May 29, 2018 by Teachoo
Transcript
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
Proving True - If and only if
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