Question

Which of the following sets are closed under division? Select all that apply.

integers
irrational numbers
whole numbers
polynomials
none of the above

Answer 1

The appropriate choice is probably
none of the above


While the inverse of any irrational number is irrational, their ratio my not be, for example (√8)/(√2) is rational.

Answer 2

None of the Above

Explanation:

Integers:

Take
`a=1,b=2`

Here,
`(a)/(b)=(1)/(2)`

Since
`(1)/(2)` is not an integer , integers are not closed under division .

Irrational Numbers:

Take
`a=√(3),b=√(27)`

Here,
`(a)/(b)=(√(3) )/(√(27) )=(1)/(√(9) )=(1)/(3)`

Since
`(1)/(3)` is not an irrational number , irrational numbers are not closed under division.

Whole Numbers:

Take
`a=2,b=3`

Here,
`(a)/(b) =(2)/(3)`

Since
`(2)/(3)` is not a whole number, whole numbers are not closed under division.

Polynomials:

Take two polynomials as
`2x+3 , 0`

Here,
`(2x+3)/(0)` is not defined.

Since
`(2x+3)/(0)` is not a polynomial, polynomials are not closed under division .

So, the correct option is none of the above