Identify the quantifier in the following statements and write the negation of the statements.

**(i)** There exists a number which is equal to its square.

**(ii)** For every real number x, x is less than x + 1.

**(iii)** There exists a capital for every state in India.

Asked by Pragya Singh | 11 months ago | 89

**(i)** The quantifier is ‘There exists’.

The negation of this statement is as follows.

There does not exist a number which is equal to its square.

**(ii)** The quantifier is ‘For every’.

The negation of this statement is as follows.

There exist a real number x for which x is not less than x +1

**(iii)** The quantifier is ‘There exists’.

The negation of this statement is as follows.

There exists a state in India whose capital does not exists.

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.”