Question 1196050


Non-zero rational number sets is closed under division.
To find : Closed under division from the following sets.


(a) let’s check Natural numbers :

    Natural numbers starts from {1, 2, 3, ......., n} numbers.

   To find whether natural numbers is closed under division, consider an example :        

     Assume two values like {{{4}}} and {{{5}}}.
     Divide {{{4}}} and {{{5}}} => {{{4/5 = 0.8}}}
     Resultant {{{0.8}}} is {{{not}}} a natural number.

     So, {{{natural}}} numbers are {{{not}}}{{{ closed}}} under division.


(b) let’s check Non - Zero integers :

    Non-Zero integers which have positive and negative values.

     To find whether non-zero integer is closed under division, consider an example :  

    Assume two non-zero values {{{2}}} and{{{ 3}}}.
    Divide them, which gives {{{2/3 = 0.6}}}
    Resultant value is{{{ 0.6}}} which is{{{ not}}} an {{{integer}}}.

    That is, non-zero {{{integers }}}are {{{not}}}{{{ closed}}} under division


(c) let’s check  Irrational numbers :

     Irrational numbers, a number cannot be represented as fraction.

     Eg : {{{sqrt(2), {{{sqrt(3)

     To find whether irrational numbers is closed under division, consider an example :  

      Assume two same irrational numbers {{{sqrt(3)}}}  and {{{sqrt(3)}}}
      Divide the values, gives {{{sqrt(3)/sqrt(3) = 1}}}.
      " {{{1 }}}", which is {{{not}}} an {{{irrational}}} numbers.

      So, {{{irrational}}} numbers are {{{not}}} {{{closed}}} under division.



(d) let’s check  Non - Zero rational numbers :

     Non zero rational numbers like {{{1/2}}}, {{{1/3}}}, {{{1/4}}}, {{{1/5 }}}and so on.

     To find whether non zero rational numbers is closed under division, consider an example :

      Assume two non zero rational numbers {{{1/6}}} and {{{ 1/7}}}.
      Dividing them we get,   {{{(1/6)/( 1/7)=7/6}}}.
      Resultant value  is the {{{rational}}} numbers.
      Hence, non zero {{{rational}}} {{{numbers}}}{{{ are}}} closed}}} under division.


Therefore, non-zero{{{ highlight(rational)}}} {{{highlight(numbers) }}}are {{{highlight(closed)}}} under {{{highlight(division)}}}.