**Question:**

**Identify the Quantifiers in the following statements.**

**(i) There exists a triangle which is not equilateral.**

**(ii) For all real numbers x and y, xy = y x.**

**(iii) There exists a real number which is not a rational number.**

**(iv) For every natural number x, x + 1 is also a natural number.**

**(v) For all real numbers x with x > 3, x 2 is greater than 9.**

**(vi) There exists a triangle which is not an isosceles triangle.**

**(vii) For all negative integers x, x 3 is also a negative integers.**

**(viii) There exists a statement in above statements which is not true.**

**(ix) There exists a even prime number other than 2.**

**(x) There exists a real number x such that x 2 + 1 = 0.**

**Solution:**

**(i) There exists a triangle which is not equilateral.**

Quantifiers means a phrase like ‘there exist’, ’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a triangle which is not equilateral”

Quantifier is “There exist”

Hence, There exist is quantifier.

**(ii) For all real numbers x and y, xy = y x.**

Quantifiers means a phrase like ‘there exist’, ’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “For all real numbers x and y, xy = yx.”

Quantifier is “For all”

Hence, ‘For all’ is quantifier.

**(iii) There exists a real number which is not a rational number.**

Quantifiers means a phrase like ‘there exist’, ’for all’ and ‘for every’ etc. and these are used to make the prepositional statement.

In the given statement “There exists a real number which is not a rational number.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

**(iv) For every natural number x, x + 1 is also a natural number.**

In the given statement “For every natural number x, x + 1 is also a natural number.”

Quantifier is “For every”

Hence, ‘For every’ is quantifier.

**(v) For all real numbers x with x > 3, x 2 is greater than 9.**

In the given statement “For all real numbers x with x > 3, x2 is greater than 9.”

Quantifier is “For all”

Hence, ‘For all’ is quantifier.

**(vi) There exists a triangle which is not an isosceles triangle.**

In the given statement “There exists a triangle which is not an isosceles triangle.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

**(vii) For all negative integers x, x 3 is also a negative integers.**

In the given statement “For all negative integers x, x3 is also a negative integers.”

Quantifier is “For all”

Hence, ‘For all’ is quantifier.

**(viii) There exists a statement in above statements which is not true.**

In the given statement “There exists a statement in above statements which is not true.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

**(ix) There exists a even prime number other than 2.**

In the given statement “There exists a even prime number other than 2.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.

**(x) There exists a real number x such that x 2 + 1 = 0.**

In the given statement “There exists a real number x such that x2 + 1 = 0.”

Quantifier is “There exist”

Hence, ‘There exist’ is quantifier.