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

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

Write the negation of each of the following statements: p: All birds have wings. q: For every real number x, x^(2) gt x. r: There exists a real number x such that 0 lt x lt 1. s: For every real number x, either x gt 1 or x lt 1.

Write the negation of the statement : p : There exist a number x such that 0 lt x lt 1 is "____________________" .

Write the negation of the following statement: r: There exists a number x such that ' 0

Identify the quantifier in the following statement and write the negation of the statement (i) p : For every negative number x,x-2 is smaller than x (ii) q : There exists a rational number whose sqaure is smaller than the number (iii) r : For every real number , there is a position on number line

Let S be a non-empty subset of R. Consider the following statement: P: There is a rational number x in S such that x"">""0 . Which of the following statements is the negation of the statement P ? There is no rational number x in S such that xlt=0 (9) Every rational number x in S satisfies xlt=0 (18) x in S and xlt=0=>x (27) is not rational There is a rational number x in S such that xlt=0 (36)

Let p be the statement “x is an irrational number,' q be the statement 'y is a trascendental number', and r be the statement 'x is a rational number iff y is a transcendental number '. Statement-1 r is equivalent to either q or p. Statement-2:r is equivalent to (p harr~q)