Q. 25.0( 3 Votes )

State whether the following statements are true or false:

(a) The set of all even natural numbers is a smallest element.

(b) Every non-empty subset of has the smallest element.

(c) Every integer can be identified with a rational number.

(d) For each rational number, one can find the next rational number.

(e) There is the largest rational number.

(f) Every integer is either even or odd.

(g) Between any two rational numbers, there is an integer.

Answer :

(a) The set of all even natural numbers is a smallest element – False


The set of all even natural number consist of elements (2, 4, 6, 8,10,……) but the smallest element is 0 which is not included in the set of natural numbers and the smallest natural number is 1 which is also not included in set even numbers.


(b) Every non-empty subset of has the smallest element. – true


Every non empty subset of Z have the smallest element. It depends on the conditions attached to the non-empty subset such as the set of even natural numbers has 2 as its least element. The set of prime numbers bigger than 5 has 7 as its least element


(c ) Every integer can be identified with a rational number- true


Every integer is a rational number as each integer n can be written in the form .


For example 7 = 7/1 and thus 7 is a rational number.


(d) For each rational number, one can find the next rational number.- False


One cannot always find the next rational number for each rational number as there also lies irrational numbers in-between.


(e) There is the largest rational number.- False


The largest rational number cannot be determined as there are infinites numbers and each number can be expressed as a rational number.


(f) Every integer is either even or odd – true


We know n is even if n = 2k for some integer k and n is odd if n = 2k + 1 for some integer k.


Every integer can be expressed as 2k or 2k + 1 hence every integer is either odd or even.


(g) Between any two rational numbers, there is an integer. - false


Rational number can be any number 5/1, 2/1, which can be expressed in form of fraction and integers are the whole numbers which are not a fraction so there cannot lie a whole number between two rational number which are fractions.


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