Write the negation of the following statements:

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

**(ii)** q: All cats scratch.

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

**(iv)** s: There exists a number x such that 0 < x < 1.

Asked by Pragya Singh | 1 year ago | 109

**(i)** The negation of statement p is as follows.

There exists a positive real number x, such that x -1 is negative

**(ii)** The negation of statement q is as follows.

There exists a cat that does not scratch.

**(iii)** The negation of statement r is as follows.

There exists a real number x , such that neither x >1 nor x < 1.

**(iv)** The negation of statement s is as follows.

There does not exist a number x , such that 0 < x < 1.

Answered by Abhisek | 1 year ago

Determine 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.”