MATH135 Lecture Notes - Lecture 3: Determinacy, Mathematical Induction, Prime Number

• School: University of Waterloo
• Department: Mathematics
• Course: MATH135
• Professor: Gabrielde Morais Coutinho

Document Summary

University of waterloo - fall 2014 - math 135. X s, y t : x2 > y2 or y < x. X s : y t, x2 + y2 > 1 = xy = 2. Now check the following examples on how to disprove a statement using its negation. Statement: for every x [0, 1], x2 < 1 . To disprove it, we can actually work around and show that its negation is true. Its negation is there exists x [0, 1] such that x2 1 . Statement: there is n z such that for all m z, 7 does not divide n + m . One of the most useful proof techniques is called proof by contradiction . We already saw some examples using this technique. The idea is that you should (i) assume the opposite of what you"re trying to prove. (ii) find a logical contradiction.