Q11 of 69 Page 261

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 = yx.


(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, x2 is greater than 9.


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


(vii) For all negative integers x, x3 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 x2 + 1 = 0.

(i) 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) 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) 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) 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) 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) 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) 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) 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) 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) 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.


More from this chapter

All 69 →
9

Write down the contrapositive of the following statements:

(i) If x = y and y = 3, then x = 3.


(ii) If n is a natural number, then n is an integer.


(iii) If all three sides of a triangle are equal, then the triangle is equilateral.


(iv) If x and y are negative integers, then xy is positive.


(v) If natural number n is divisible by 6, then n is divisible by 2 and 3.


(vi) If it snows, then the weather will be cold.


(vii) If x is a real number such that 0 < x < 1, then x2 < 1.

 10

Write down the converse of following statements :

(i) If a rectangle ‘R’ is a square, then R is a rhombus.


(ii) If today is Monday, then tomorrow is Tuesday.


(iii) If you go to Agra, then you must visit Taj Mahal.


(iv) If the sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right angled.


(v) If all three angles of a triangle are equal, then the triangle is equilateral.


(vi) If x : y = 3 : 2, then 2x = 3y.


(vii) If S is a cyclic quadrilateral, then the opposite angles of S are supplementary.


(viii) If x is zero, then x is neither positive nor negative.


(ix) If two triangles are similar, then the ratio of their corresponding sides are equal.

 12

Prove by direct method that for any integer ‘n’, n3 – n is always even.

 13

Check the validity of the following statement.

p : 125 is divisible by 5 and 7.