7 of such a pair What about a 1 and b 1 They serve the purpose In fact there are infinitely many counterexamples Why Now an exercise E12 Disprove the following statements by providing a suitable counterexample i VxEZ x N ii x y n x y Vn N x y Z iii f N N is 1 1 iff f is onto Hint To disprove p q it is enough to prove that p q is false or qp is false There are some other strategies of proof like a constructive proof which you will come across in the appendix to Unit 11 and in other mathematics courses We shall not discuss this method here Other proof related adjectives that you will come across are vacuous and trivial A vacuous proof makes use of the fact that if p is false then p q is true regardless of the truth value of q So to vacuously prove p q all we need to do is to show that p is false For instance suppose we want to prove that If n n 1 for n Z then n 0 Since n n 1 is false for every ne Z the given statement is vacuously true or true by default Similarly a trivial proof of p q is one based on the fact that if q is true then p q is true regardless of the truth value of p So for example If n n 1 for n E Z then n 1 n is trivially true since n 1 nVnEZ The truth value of the hypothesis which is false in this example does not come into the picture at all Here s a chance for you to think up such proofs now E13 Give one example each of a vacuous proof and a trivial proof And now let us study a very important technique of proof for statements that are of the form p n n N 2 4 PRINCIPLE OF INDUCTION In a discussion with some students the other day one of them told me very cynically that all Indian politicians are corrupt I asked him how he had reached such a conclusion As an argument he gave me instances of several politicians all of whom were known to be corrupt What he had done was to formulate his general opinion of politicians on the basis of several particular instances This is an example of inductive logic a process of reasoning by which general rules are discovered by the observation of several individual cases Inductive reasoning is used in all the sciences including mathematics But in mathematics we use a more precise form