082nd_tut1sol - MATH1111/2008-09/Tutorial I Solution 1...

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MATH1111/2008-09/Tutorial I Solution 1 Tutorial I Suggested Solution 1. This question concerns how to prove or disprove a statement. We start with ”common sense”. For statments (a) and (b) below, think about what you need to do if you want to prove it. Also, think about how you can disprove it. (a) All apples are sweet. (b) Some apples are sweet. Prove or disprove the following. (i) All matrices X satisfying X 2 = 0 are zero matrices. (ii) Some square matrices B satisfying B 2 - I = 0 are singular. (iii) If A is a square matrix such that A 2 - I = 0, then there exists a nonzero vector x such that Ax = x or Ax = - x . Ans . (a) To prove ”All apples are sweet”, you must find a way to check through all apples and verify that all of them are sweet. To disprove it, it is in principle simple. Once you find an apple which is not sweet, then you are done. This approach is disproof by counterexample . (b) To prove ”Some apples are sweet”, now we can simply prove it by finding a sweet apple. (Just one is enough to verify the statement! My goal is actually to explain the difference between ”for all” and ”for some”. So don’t argue that we need to find at least two sweet apples from the plural form in the statement.)
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