Question

Question 9. True or false? Provide a simple counter-example if it is false. (9.a) The sum of two rational numbers x, y EQ is always a rational number, x+y EQ. (9.b) The sum of two irrational numbers x, y ER - Q is always an irrational number, x +y ER-Q. (9.c) The sum of the squares of two distinct real numbers x,y ER, with x #y, is always a positive real number, x2 + y2 ER and x2 + y2 > 0. (9.d) If a, b ER, then ab E R.

Answer 1

a. True. Rational numbers are closed under the sum operation, therefore, the sum of two rational numbers is always a rational number.

b. True. irrational numbers are closed under the sum operation, therefore, the sum of two irrationals numbers is always a irrational number.

c. True. The square of a real number is always a number greater than zero, and the sum of two numbers greater than zero is greater than zero.

d. True. The real numbers are closed under the product operation, then if a and b are reals numbers, the product ab is also a real number.

Explanation: