Does x<=b Imply max(x)=b and How Do Set Operations Differ?

[torquerotates]

If x<=b does this mean max(x)=b?

is x<=b equivalent to the interval (-infinity, b]?

 

[mathwonk]

i think so.

 

[Goongyae]

>If x<=b does this mean max(x)=b?

No.
3 <= 4, but max(3) is not 4. It's only 3.

>is x<=b equivalent to the interval (-infinity, b]?

Yes, assuming x and b can't themselves be -infinity.

 

[mathman]

If x<=b does this mean max(x)=b?

is x<=b equivalent to the interval (-infinity, b]?

You need to be more specific about x and b. Is x a constant or an unknown or a variable? b should also be defined as to what kind of thing it is.

 

[torquerotates]

@math man, x is an unknown while b is a constant. The thing that is confusing me is that my analysis textbook defined the sup of a set, call it S, as having the following property, if s is in S, then s<=sup(S). Now that is confusing bc they didn't specify if sup(S) belonged to S. But s<=sup(S) means that s is in the interval (-infinity, sup(S)] implying max(S)=sup(S). Its as if the definition of supremum is forcing sup(S) to be the max of S.

So the only way I around it is that if sup(S) not in S, then s<sup(S) => s<=sup(S). Which makes sense because if s is strictly less than sup(S), I think I can say that it is strictly less than or equal to sup(S). Is this line of thinking correct?

 

[l'Hôpital]

What they are trying to get at is the following:

Consider the set S = [0,1). What is the sup? Does it have a maximum?

 

[torquerotates]

The sup is 1 and it has no max. But does s<1 mean s<=1?

 

[Goongyae]

does s<1 mean s<=1?

Yes. s<1 also means s<2000 and s<=2000.

defined the sup of a set, call it S, as having the following property, if s is in S, then s<=sup(S).

That's a crappy definition. Because then the sup of {1,2,3,4} could be 4, 4.5, 5, or 5000.
The supremum is normally defined as the smallest such possible value; in my example, 4.

 

[torquerotates]

Sorry I forgot to include that if e>0 then there exists s* s.t
supS-e<s*<=supS.

 

[torquerotates]

wait, so if a<b => a<=b, then why does the same logic not work for sets? for example, why isn't A=AorB. Looking at a venn diagram, we clearly see that the area of the union is greater than the area of A.