 In this question, we have to prove the validity of the given statements by a contradiction method. So we should first assume that the statement p is not true, that is, negation of p is true, and then proceeding with the question, we should arrive at some result which contradicts the hypothesis. Therefore, we conclude that negation of p is not true or p is true. So the method of contradiction says that if p is any statement, then to prove its validity, suppose negation of p is true and then arrive at a contradiction. So keeping this in mind, let's now begin with the solution. The given statement is the sum of an irrational number and a rational number is irrational. Now let if possible, negation of p is true, that is, the sum is, is a rational number. X to be a rational number, root of y, an irrational number. Therefore root of y is equal to z, where z belongs to q. Now this implies square root of y is equal to z minus x. Now z minus x is rational by the closure and property of subtraction in rational numbers. Square root of y is an irrational number by our assumption y equals to z minus x means that a rational number is equal to irrational number which is not possible. The supposition that negation of p is true is wrong, hence the statement p is true. So this completes the first part. Let's now move on to the second part. Second statement is if n is a real number where n is greater than 3, then n squared is greater than 9. A contradiction method. Let's now begin with the solution. If possible, let negation of q is true, that is, for n greater than 3, n squared is less than 9 where n belongs to r. Since this is greater than 3, therefore let n is equal to 3 plus x where x belongs to n. Now n equals to 3 plus x in height, n squared is equal to 9 plus 6x plus x squared. Now n squared cannot be less than 9x plus x squared is greater than 0 for x belonging to n. Hence our supposition, negation of q is true is wrong, hence the statement q is true. So we have checked the validity of both the statements by the contradiction method. So this completes the session. Bye and take care.