Solution Let us start by assuming A is a truth teller What A
Last updated: 10/12/2023
Solution Let us start by assuming A is a truth teller What A says is true B is a truth teller What B says is true A and B are of opposite types This is a contradiction because our premises say that A and B are both truth tellers The assumption we started with is false A always tells lies What A has told you is a lie B always tells lies A and B are of the same type i e both of them always lie Here are a few exercises for you now While doing them you would realise that there are situations in which all the three methods of proof we have discussed so far can be used E9 Use the method of proof by contradiction to show that i 3 is irrational ii For x E R if x 4x 0 then x 0 6 E10 Prove E 9 ii directly as well as by the method of contrapositive E11 Suppose you are visiting the village described in Example 7 above Another two villagers C and D approach you C tells you Both of us always tell the trutli and D says C always lies What types are C and D Let us now consider the problem of showing that a statement is falsc 2 3 3 Counterexamples Suppose I make the statement All human beings are 5 feet tall You are quite likely to show me an example of a human being standing nearby for whom the statement is not true And as you know the moment we have even one example for which the statement Vx p x is false i e 3x p x is true then the statement is false An example that shows that a statement is false is a counterexample to such a statement The name itself suggests that it is an example to counter a given statement A common situation in which we look for counterexamples is to disprove statements of the form p q From Unit 1 you know that p q PA q Therefore a counterexample to p where p q is true i e p is true and holds but the conclusion q does not hold q needs to be an example q is true i e the hypothesis p For instance to disprove the statement If n is an odd integer then n is prime we need to look for an odd integer which is not a prime number 15 is one such integer So n 15 is a counterexample to the given statement Notice that a counterexample to a statement p proves that P is false i e p is true Let us consider another example Example 8 Disprove the following statement VaR VbER a b a b