Exploring Unbounded and Bounded Sets

[BelaTalbot]

Is a bounded set synonymous to a set that goes to infinity? I feel like unless a set is
(-infinity, n) or [n, infinity) it is not going to be unbounded.

The other thing that I was wondering is can a set be neither open nor closed AND unbounded? Doesn't the definition of open/closed imply that there is a boundary?

Thanks!

 

[adriank]

A set being bounded just means it is contained in some ball. Boundedness isn't really related to being open or closed, and also has nothing to do with boundary.

 

[Studiot]

Is a bounded set synonymous to a set that goes to infinity?

No and furthermore the bound does not even have to be a member of the set.

A set can have lots of bounds, even an infinite number of them.

A bound can be 'above' or 'below'.

An upper bound is simply a real number that is either greater than or equal to every member of the set or less than/equalto every member of the set.

so 7,8,9,10 etc all form upper bounds to the set {2,3,4,5,6} ; none are memebrs of the set.
but wait 6 also forms an upper bound as it satisfies the equal to and is a member.

Similarly 1,0,-1,-2 all form lower bounds that are not members and 2 forms a lower bound that is

Sets which 'go to infinity' are unbounded. However a set can contain an infinite number of members and still be bounded, above and/or below.

for example the set {1/1, 1/2, 1/3, ...} is bounded above by 2 ,1 etc and bounded below by 0, -1 etc, but contains an infinite number of members.

The set $$\{ - \infty ,... - 2, - 1,0,1,\frac{1}{2},\frac{1}{3},...\}$$

is not bounded below but is bounded above.

If a set has both a lower and upper bound so that the modulus of any member, x, is less than or equal to some real number K then the set is bounded. (No upper or lower)

If for any $$x \in S$$ there exists a K such that

$$\left| x \right| \le K$$

The set S is bounded.

Hope this helps.

 

[g_edgar]

"bounded" does not mean "has a boundary"

Isn't English confusing?

 

[BelaTalbot]

So I guess what I'm really wondering is can a set be unbounded if it doesn't go to infinity? That's what I can't seem to wrap my brain around, because I feel that if a set doesn't go to infinity, there will always be a real number K larger than the members of the set.

 

[Studiot]

Any finite set S of real numbers is bounded.
Infinite sets can be bounded or unbounded.

Remember that to be bounded a set must have both a lower bound and an upper bound.
Sets with only a lower or upper bound are unbounded.

A further question for you to ponder:

Can a set of complex numbers be bounded?