What are upper and lower bounds and why are they important in mathematics?

[woundedtiger4]

At:

http://en.wikipedia.org/wiki/Upper_and_lower_bounds

in example it says that

"2 and 5 are both lower bounds for the set { 5, 10, 34, 13934 }, but 8 is not"

Why "2"? as 2 is not in that set.

Also,
at:

http://en.wikipedia.org/wiki/Supremum

in example it says that
"The "Supremum" or "Least Upper Bound" of the set of numbers 1, 2, 3 is 3. Although 4 is also an upper bound, it is not the "least upper bound" and hence not the "Supremum"."

Why? as 4 is not in the set of 1,2,3 but if for a moment I think that as 4>3 so it is the upper bound of the set which contains 1,2,3 then am I correct to say that 3 is the least upper bound ?

Thanks in advance.

 

[micromass]

A lower bound of a set A is any number x such that x<a for any $a\in A$.
So we don't need the number to be in the set (if the element is in the set, then it's called a minimum). A lower bound is just any number smaller than each element in the set.

 

[woundedtiger4]

A lower bound of a set A is any number x such that x<a for any $a\in A$.
So we don't need the number to be in the set (if the element is in the set, then it's called a minimum). A lower bound is just any number smaller than each element in the set.

So 5 is the greatest lower bound.

 

[micromass]

Yes. Anything lower than 5 is also a lower bound.

 

[woundedtiger4]

(Copy/pasted from wiki)

Question: as Q doesn't have the least upper bound as 2^1/2 is irrational but the example says that Q has an upper bound, is that upper bound any number greater than 2^1/2 or is it a specific number?

 

[woundedtiger4]

Yes. Anything lower than 5 is also a lower bound.

Thank you so much.

 

[micromass]

View attachment 59566

(Copy/pasted from wiki)

Question: as Q doesn't have the least upper bound as 2^1/2 is irrational but the example says that Q has an upper bound, is that upper bound any number greater than 2^1/2 or is it a specific number?

Any rational number greater than $\sqrt{2}$ is an upper bound.

 

[woundedtiger4]

Any rational number greater than $\sqrt{2}$ is an upper bound.

Once again, thank you very much sir.

 

[HallsofIvy]

So 5 is the greatest lower bound.

In fact, because 5 is in the set, 5 is the minimum of the set.
(If a set has a minimum (smallest member) then that minimum is the greatest lower bound.) But as long as a set has lower bounds, it has a greatest lower bound whether is has a minimum or not.

 

[micromass]

But as long as a set has lower bounds, it has a greatest lower bound whether is has a minimum or not.

Of course, that is only true in $\mathbb{R}$.

 

[HallsofIvy]

View attachment 59566

(Copy/pasted from wiki)

Question: as Q doesn't have the least upper bound as 2^1/2 is irrational but the example says that Q has an upper bound, is that upper bound any number greater than 2^1/2 or is it a specific number?

No, it does not say that Q has an upper bound! It says that Q intersect the interval from $$-\sqrt{2}$$ to $$\sqrt{2}$$ has upper bounds. 1.5, for example is an upper bound of that set.

(But Q, the set of all rational numbers, does NOT have either upper or lower bounds.)

 

[woundedtiger4]

What does upper bound, least upper bound (supremum), lower bound, and greatest lower bound (infimum) tells us intutively? It just tells us lower & greater numbers, right?

Is maximum (max) is just an other word for least upper bound (supremum), and similarly minimum (min) is just an other word for greatest lower bound (infimum)?

 

[micromass]

What does upper bound, least upper bound (supremum), lower bound, and greatest lower bound (infimum) tells us intutively? It just tells us lower & greater numbers, right?

Yes, the upper and lower bound just tells us lower and greater numbers. The greatest lower bound also tells us a lower number, but the best possible one.

Is maximum (max) is just an other word for least upper bound (supremum), and similarly minimum (min) is just an other word for greatest lower bound (infimum)?

Not exactly. A minimum of a set A is an infimum that also belongs to the set.
For example, 1 is an infimum of (1,2], but not a minimum since 1 does not belong to the set. On the other hand, 1 is a minimum of [1,2] and thus also an infimum.