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Transcript
Methods of proving theorems â Counterexample 21 ï± Counterexample : To show that a statement of the form âxP(x) is false, we need only find a counterexample, that is, an example x for which P(x) is false. ï E.g. Prove that âFor all real numbers x2 > xâ is false X=0.5 is a counterexample since 0.52 > 0.5 is not true ï± Example Show that the statement: âEvery positive integer is the sum of the squares of two integersâ is false. ï To show that this statement is false, we look for a counterexample which is 3 because we cannot write 3 as the sum of the squares of two integers. To show this is the case, note that the only perfect squares not exceeding 3 are: (02 = 0 & 12 = 1), but 3 ï¹ 1+0. Therefore the statement is false
