Question

Which of the following sets is closed under division?
natural numbers
non-zero integers
irrational numbers
non-zero rational number

Answer 1

non-zero rational numbers because because it is impossible to divide our way out of the set of nonzero rational numbers. For example we can start with two nonzero rational numbers, say  and , which is indeed a nonzero rational number. So we cannot divide our way out of the set of nonzero rational numbers. Since this is not possible, the set of nonzero rational numbers is indeed closed under division.

Answer 2

Non-zero rational number sets is closed under division.

To find :

Closed under division fro the following sets.

Given :

(a) 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) 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) Irrational nubmers :

     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 irrational numbers $\sqrt{4}$ and
•       Divide the values, gives $\frac{\sqrt{4} }{\sqrt{4} }$ = 1.
•       " 1 ", which is not an irrational numbers.
•       So, irrational numbers are not closed under division.

(d) 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,   $\frac{1}{6}$ ÷ $\frac{1}{7}$ $= \frac{1}{6}$ × $\frac{7}{1} = \frac{7}{6}$.
•       Resultant value $\frac{7}{6}$ is the rational numbers.
•       Hence, non zero rational numbers are closed under division.

Therefore, non-zero rational numbers are closed under division.

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