Example 14 - Mathematical Reasoning
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 14 Check whether the following statement is true or false by proving its contrapositive. If x, y ∈ Ζ such that xy is odd, then both x and y are odd. Let p : xy is odd. q : both x and y are odd. The given statement is of the form it p then q i.e. p ⇒ q We have to check the validity of statement by contrapositive method. i.e. x and y are not odd & prove that is false i.e. prove that xy is not odd Now consider x & y are not odd i.e. x & y are even let x = 2n for same n, m ∈ Z & t = 2m Calculate xy = 2n × 2m if n, m ∈ Z = 4mn ⇒ nm ∈ Z = 2 (2 m n) ⇒ 2n m ∈ Z This show xy is even ⇒ xy is not odd ⇒ p is false ⇒ ~ p Hence ~ q ⇒ ~ p
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About the Author
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 and Computer Science at Teachoo