1. ## Empty set

What can we say about an empty set? Is it bounded below? Bounded above?

You can't really contradict the fact that it's NOT bounded below/above, but you can't really prove it either? So it's not right to say that it's bounded below/ above, or it's not bounded below/above?

Is my logic correct?
Thank you.

2. ## Re: Empty set

Since the empty set has no members, when it is considered as a subset of any ordered set, then every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set.

Empty set - Wikipedia, the free encyclopedia

3. ## Re: Empty set

Originally Posted by davidciprut
What can we say about an empty set? Is it bounded below? Bounded above?
Thank you.
You can't say anything. Bounds are not defined for an empty set.
If y>=x for all x in S, y is an upper bound. There are no x in the empty set.

But there is a point of logic here, which I never liked, and which may be wrong:
There is no point in S for which x>y. Therefore y must be an upper bound. But if y is an upper bound, y>=x for all x in S. There is something wrong here.

You can take me to school on this.

4. ## Re: Empty set

Missed the "edit." I note wiki begins its definition of bound on S with the expicit specification that S be non-empty. That avoids the ambiguous logic problem of the previous post.

Least-upper-bound property - Wikipedia, the free encyclopedia

5. ## Re: Empty set

Originally Posted by Hartlw
Missed the "edit." I note wiki begins its definition of bound on S with the expicit specification that S be non-empty. That avoids the ambiguous logic problem of the previous post.

Least-upper-bound property - Wikipedia, the free encyclopedia
You posted a link for "Least upper bound property", which does not give the definition for a bound on S. It gives the definition for a Least-upper bound on S. Not every upper bound is a least-upper bound. Not every lower bound is a greatest-lower bound. The definitions for upper bounds and lower bounds do not require that the subset be non-empty:

Upper and lower bounds - Wikipedia, the free encyclopedia

The definition states that if $\displaystyle S$ is a subset of the poset $\displaystyle (K,\le)$, then an element $\displaystyle x\in K$ is an upper bound if for all $\displaystyle y \in S, y\le x$. It does not require the existence of $\displaystyle y \in S$. It does require the existence of $\displaystyle x \in K$. Example: as a subset of the reals, 0 is an upper bound of the empty set since given any element of the empty set, 0 will be greater than or equal to it. That condition is trivially met as there are no elements of the empty set with which to compare it.

6. ## Re: Empty set

Originally Posted by SlipEternal
Upper and lower bounds - Wikipedia, the free encyclopedia
The definition states that if $\displaystyle S$ is a subset of the poset $\displaystyle (K,\le)$, then an element $\displaystyle x\in K$ is an upper bound if for all $\displaystyle y \in S, y\le x$.
Also it is a simple matter of logic: a false statement implies any statement.

If $\displaystyle x \in \emptyset$ then $\displaystyle x \le 1$ is a true statement

If $\displaystyle x \in \emptyset$ then $\displaystyle x \ge 1$ is a true statement.

7. ## Re: Empty set

Originally Posted by SlipEternal
You posted a link for "Least upper bound property", which does not give the definition for a bound on S.
Upper and lower bounds - Wikipedia, the free encyclopedia

"Let S be a non-empty set of real numbers.
A real number x is called an upper bound for S if x ≥ s for all s ∈ S.
A real number x is the least upper bound (or supremum) for S if x is an upper bound for S and x ≤ y for every upper bound y of S." underlining added.

8. ## Re: Empty set

This is sort of like asking:

"Where is the empty set on the real line"?

On the one hand, it's EVERYWHERE, because it is contained in ANY interval (a,b).

On the other hand, it's NOWHERE, because no real number at any point on the line is an element.

If you have a blank piece of paper on which you have yet to draw a line, where is the line?

That is the logical dilemma of the empty set: it doesn't really make "that much sense". Fortunately, there's nothing in it, so we don't really have to worry about it.

9. ## Re: Empty set

There is no logical dilemma:

Upper bound only defined for a non-empty set is not a logical dilemma.

10. ## Re: Empty set

I did not say *that* was the logical dilemma, rather it is that the empty set has all sorts of contradictory properties. It doesn't make sense to ask: is the empty set bounded? It is, and it isn't. It's both, it's neither. For example, as an interval, it both doesn't and does contain its endpoints (it is both open and closed).

More explicitly, if P is a proposition concerning a subset of the real numbers, then Ø = {A: P(A) and not P(A)}, including (for example) P(A) = "A is a bounded set".

Every property regarding sets is vacuously true of the empty set: unfortunately (or perhaps VERY fortunately), this isn't saying much.

11. ## Re: Empty set

Deveno, why do you keep harping on the empty set? It has nothing to do with this thread. Bounds are not defined for empty sets. That's the answer to the OP. If you wish to philosphize about the empty set, start another thread.

12. ## Re: Empty set

Um...it's in the thread title? The thread starter wished to know whether the empty set has certain properties. He asked if his logic was correct...I am merely trying to answer that question. My answer is not that much different in CONTENT than post #6.

Post #5 specifically addresses the point that a definition of a bound need not require the set is non-empty. Bounds of empty sets are, however, not particularly enlightening to consider (we don't learn anything, every element works).

Yes, one MAY take the approach that one will consider properties that only are applied to non-empty sets. In some cases, this leads to less confusion at the (small) expense of a few more words. I humbly submit this is not the ONLY possible approach.