For any positive real number `x`, prove that there exists an irrational number `y` such that `0 lt y lt x`.

If x and y are positive real number, then :

Show that the statement : "Given a positive number x, there exists a rational number r such that 0 lt r lt x is true.

If x and y are positive real numbers , then prove that x lt y iff x^2 lt y^2

For rational numbers x and y, if x lt y then x - y is a positive rational number.

Write the negation of the following statements s : There exists a number x such that 0 lt x lt 1

Assertion : One of the rational number between (1)/(5) and (1)/(4) is (9)/(2) . Reason : If x and y are any two rational numbers such that x lt y , then (1)/(2)(x+y) is a rational number between x and y such that x lt (1)/(2) (x+y) lt y .