Question

why set of irrational nos are not dense??? while we consider this def of dense set
a set A is said to be dense in R if there is a point of A between any two real nos....

Answer 1

Does the definition specify that the two real numbers are distinct? After all, there is no irrational number "between" two indistinct rational numbers...

Answer 2

e.g???

Answer 3

Well, between 1 and 1 for example.

Answer 4

yes rel nos must be distinct

Answer 5

but these r not distinct

Answer 6

Hmm. Between any two real distinct numbers there is indeed always an irrational number (and a rational one for that matter).

So that definition is not very good...

Answer 7

but i want the ans of my que according to this def???

Answer 8

If you insist on using this definition (and specifying that the real numbers you choose must be distinct), then the irrationals are "dense".
Where does this question/definition com from? Is this the exact wording?

Answer 9

@rosy :May I join ?
@Erin001001

Answer 10

@E.ali Of course :-) You don't need to ask, just join in!

Answer 11

K !
Thank you !
So whats the problem ???

Answer 12

yes this is exact def....alternatively we can say every open interval of R contains a pt of A

Answer 13

yes why not E.ali

Answer 14

you can easily see what is the problem E.ali:)

Answer 15

K !
But whats your problem in @Erin001001 ' s answer ???:)

Answer 16

set of irrational nos can never be dense in R

Answer 17

this is the problem

Answer 18

Is it your mean ?:
I cant be in R you say ...
OK ?!

Answer 19

@rosy By both definitions, the set of irrationals is dense, so they're probably not very good definitions. Regarding the second definition, the empty set is strictly speaking an open interval, so according to that definition alone (and no other assumptions), no sets would be dense...

Answer 20

Bad definitions aside, I don't really see a problem here. Why do you insist on the set of irrational not being dense?!

Answer 21

bcz it is a fact that set of rational nos is dense but irrationals are not...:(

Answer 22

Actually, the set of irrationals is dense ;-)

Answer 23

no it is wrong

Answer 24

!

Answer 25

I know this question is closed, but perhaps this can convince you:
http://www.proofwiki.org/wiki/Irrationals_Dense_in_Reals

You can find various other proofs if you google "irrationals dense". The Wikipedia article on "Dense set" also lists the set of rationals, as well as the set of irrationals, as dense sets.