# Infimum and supremum?

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#1
I keep coming along this terms in a few proofs I have been reading but I don't know what it means.

E.g.

Could some one explain what the inf and sup are, I haven't covered it in lectures which seems weird but I want to understand what it means so I thought I'd ask here to try and get a straightforward explanation.

Thanks.
0
6 years ago
#2
The supremum is the smallest number, x, such that for every s in S, s<=x. It is the smallest upper bound of the set (provided an upper bound exists).

For example the supremum of (a,b) is b. The supremum of [a,b] is b, the supremum of the natural numbers does not exist.
0
6 years ago
#3
(Original post by alex2100x)
Could some one explain what the inf and sup are, I haven't covered it in lectures which seems weird but I want to understand what it means so I thought I'd ask here to try and get a straightforward explanation.
A supremum is the generalisation of the idea of a maximum element of a set. Consider the set

Then 1 is the maximum of since any element is less than or equal to 1, and any number greater than 1 isn't in the set. Note that 1 is in the set.

Now consider the set .

Then 1 isn't the maximum of , and in fact the set has no maximum element, since for any but we can also find some i.e. given some close to 1, we can always find another element which is a bit closer to 1. Note that 1 is not in the set, so it cannot be a maximum element.

However, even if 1 isn't the maximum, it still plays a significant role in determining the "top" of the set since if we pick any number at all smaller than 1 (e.g. m = 0.9999999999 <a googleplex 9s omitted> 9), we can find some .

That says that 1 is an upper bound for (or to put it another way, that is bounded above)

2, 3, 100, etc are also upper bounds for , but 1 is a special upper bound, because it is the smallest one; it is the least upper bound for , since as I noted above, any number smaller than 1 *isn't* an upper bound.

Another name for the least upper bound of a set is the supremum of the set. So .

All subsets of that are bounded above have a supremum that is also in . This is called the least upper bound property, and it says, more or less, that is complete, that there aren't any missing points in the number line without a number to fill them.

Some sets don't have a supremum though. Consider . Now is bounded above (10 is an upper bound for example) but there is no least upper bound . The obvious candidate is , but , so there is no supremum.

That tells us that is not complete, that it doesn't have the least upper bound property, that there *are* missing gaps in the number line of .

If a supremum is also a member of a set, then it is also called the maximum of the set. So with the definition of above,

And the infimum is the same, except it generalises the role of minimum.
4
6 years ago
#4
(Original post by atsruser)
A supremum is the generalisation of the idea of a maximum element of a set. Consider the set

Then 1 is the maximum of since any element is less than or equal to 1, and any number greater than 1 isn't in the set. Note that 1 is in the set.

Now consider the set .

Then 1 isn't the maximum of since for any but we can also find some i.e. given some close to 1, we can always find another element which is a bit closer to 1. Note that 1 is not in the set.

However, even if 1 isn't the maximum, it still plays a significant role in determining the "top" of the set since if we pick any number at all smaller than 1 (e.g. m = 0.9999999999 <a googleplex 9s omitted> 9), we can find some .

That says that 1 is an upper bound for (or to put it another way, that is bounded above)

2, 3, 100, etc are also upper bounds for , but 1 is a special upper bound, because it is the smallest one; it is the least upper bound for , since as I noted above, any number smaller than 1 *isn't* an upper bound.

Another name for the least upper bound of a set is the supremum of the set. So .

All subsets of that are bounded above have a supremum that is also in . This is called the least upper bound property, and it says, more or less, that is complete, that there aren't any missing points in the number line without a number to fill them.

Some sets don't have a supremum though. Consider . Now is bounded above (10 is an upper bound for example) but there is no least upper bound . The obvious candidate is , but , so there is no supremum.

That tells us that is not complete, that it doesn't have the least upper bound property, that there *are* missing gaps in the number line of .

If a supremum is also a member of a set, then it is also called the maximum of the set. So with the definition of above,

And the infimum is the same, except it generalises the role of minimum.

must definitely rep this effort and time ...
(unless you are a program)
0
6 years ago
#5
(Original post by atsruser)
<snip>
This is a brilliant explanation.
0
6 years ago
#6
(Original post by alex2100x)
I keep coming along this terms in a few proofs I have been reading but I don't know what it means.

E.g.

Could some one explain what the inf and sup are, I haven't covered it in lectures which seems weird but I want to understand what it means so I thought I'd ask here to try and get a straightforward explanation.

Thanks.
(Original post by james22)
The supremum is the smallest number, x, such that for every s in S, s<=x. It is the smallest upper bound of the set (provided an upper bound exists).

For example the supremum of (a,b) is b. The supremum of [a,b] is b, the supremum of the natural numbers does not exist.
(Original post by atsruser)
A supremum is the generalisation of the idea of a maximum element of a set. Consider the set

Then 1 is the maximum of since any element is less than or equal to 1, and any number greater than 1 isn't in the set. Note that 1 is in the set.

Now consider the set .

Then 1 isn't the maximum of since for any but we can also find some i.e. given some close to 1, we can always find another element which is a bit closer to 1. Note that 1 is not in the set.

However, even if 1 isn't the maximum, it still plays a significant role in determining the "top" of the set since if we pick any number at all smaller than 1 (e.g. m = 0.9999999999 <a googleplex 9s omitted> 9), we can find some .

That says that 1 is an upper bound for (or to put it another way, that is bounded above)

2, 3, 100, etc are also upper bounds for , but 1 is a special upper bound, because it is the smallest one; it is the least upper bound for , since as I noted above, any number smaller than 1 *isn't* an upper bound.

Another name for the least upper bound of a set is the supremum of the set. So .

All subsets of that are bounded above have a supremum that is also in . This is called the least upper bound property, and it says, more or less, that is complete, that there aren't any missing points in the number line without a number to fill them.

Some sets don't have a supremum though. Consider . Now is bounded above (10 is an upper bound for example) but there is no least upper bound . The obvious candidate is , but , so there is no supremum.

That tells us that is not complete, that it doesn't have the least upper bound property, that there *are* missing gaps in the number line of .

If a supremum is also a member of a set, then it is also called the maximum of the set. So with the definition of above,

And the infimum is the same, except it generalises the role of minimum.
this is easy and i have just done gcse maths
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