a . This implies that is a common factor of both of a and b which contradicts our earlier assumption that a and b have no common factors. This shows that the assumption is rational is wrong. Hence, the statement is irrational is true.
Next, we shall discuss a method by which we may show that a statement is false. The method involves giving an example of a situation where the statement is not valid. Such an example is called a counter example . The name itself suggests that this is an example to counter the given statement.
Example By giving a counter example, show that the following statement is false. If n is an odd integer, then n is prime. Solution The given statement is in the form “if p then q ” we have to show that this is false. For this purpose we need to show that if p then ∼ q .
To show this we look for an odd integer n which is not a prime number. is one such number. So n = is a counter example. Thus, we conclude that the given statement is false.
In the above, we have discussed some techniques for checking whether a statement is true or not. A Note In mathematics, counter examples are used to disprove the statement. However, generating examples in favour of a statement do not provide validity of the statement. EXERCISE .
. Show that the statement p: “If x is a real number such that x + x = , then x is ” is true by (i) direct method, (ii) method of contradiction, (iii) method of contrapositive . Show that the statement “For any real numbers a and b , a = b implies that a = b ” is not true by giving a counter-example. .
Show that the following statement is true by the method of contrapositive. p : If x is an integer and x is even, then x is also even. . By giving a counter example,