what is dense set?

Question

what is dense set?

anonymous · Answer

if any one is good in metric spaces then plz help me

kinggeorge · Answer

The rational numbers in the real numbers is the usual example here.

anonymous · Answer

how can u explain?

anonymous · Answer

please explain it i have no idea about the dense and its example

amistre64 · Answer

dense to me would indicate tightly packed

anonymous · Answer

how can u say rational numbers in the real numbers is the usual example

anonymous · Answer

why not natural nos or whole nos.?

amistre64 · Answer

i just tried reading the definition of a dense set ... i got no idea what is talking about :/

kinggeorge · Answer

It's just the most common example given to me. It works because for any epsilon greater than 0, there is an infinite amount of rational numbers within the interval $$x-\epsilon \leq x\leq x+\epsilon $$where x is a rational number.

kinggeorge · Answer

This captures the essence of a dense set. If it is a dense set, a similar inequality will hold.

In the natural numbers, If we let $\epsilon =.5$ and $x=4$, then there is exactly one natural number in the interval $$3.5 <4<4.5$$Namely, 4.

beginnersmind · Answer

@amistre64
"i just tried reading the definition of a dense set ... i got no idea what is talking about :/"

Wikipedia gives a topological definition, which is confusing if you haven't study basic topology or real analysis. The metric picture is much clearer. It just says that Set 2 is dense in Set 1 if for any element in Set 1 and arbitrarily small distance d you can find an element of Set 2.

beginnersmind · Answer

*studied