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.
(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.
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 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.
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 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.
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 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.
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 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.
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 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.
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 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.
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 x such that x2 + 1 = 0.”
Quantifier is “There exist”
Hence, ‘There exist’ is quantifier.
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