**(i) **p : For every positive real number x, the number (x – 1) is also positive.

The negation of the statement:

p: For every positive real number x, the number (x – 1) is also positive.

is ~p: There exists a positive real number x, such that the number (x – 1) is not positive.

**(ii) **q: For every real number x, either x > 1 or x < 1.

The negation of the statement:

q: For every real number x, either x > 1 or x < 1.

is ~q: There exists a real number such that neither x>1 or x<1.

**(iii) **r: There exists a number x such that 0 < x < 1.

The negation of the statement:

r: There exists a number x such that 0 < x < 1.

is ~r: For every real number x, either x ≤ 0 or x ≥ 1.

Answered by Sakshi | 1 year agoDetermine whether the argument used to check the validity of the following statement is correct: p: “If x^{2} is irrational, then x is rational.” The statement is true because the number x^{2} = π^{2} is irrational, therefore x = π is irrational.

Which of the following statements are true and which are false? In each case give a valid reason for saying so

**(i)** p: Each radius of a circle is a chord of the circle.

**(ii) **q: The centre of a circle bisect each chord of the circle.

**(iii)** r: Circle is a particular case of an ellipse.

**(iv)** s: If x and y are integers such that x > y, then – x < – y.

**(v)** t: \( \sqrt{11}\) is a rational number.

By giving a counter example, show that the following statement is not true. p: “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.”

Show that the following statement is true “The integer n is even if and only if n^{2} is even”

Show that the following statement is true by the method of the contrapositive p: “If x is an integer and x^{2} is odd, then x is also odd.”