Infimum

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Fleshing out, make sure the caveat is known, proof of claim

Definition

Let $(X,\preceq)$ be a poset and let $A\subseteq X$ be any subset of $X$. The infimum (AKA: greatest lower bound, g.l.b) of $A$ is an element of $X$, written $\text{Inf}(A)$ that satisfies the following two conditions:

1. $\forall a\in A[\text{Inf}(A)\preceq a]$ - which states that $\text{Inf}(A)$ is a lower bound of $A$ - and
2. $\forall b\in\underbrace{\left\{x\in X\ \vert\ (\forall a\in A[x\preceq a])\right\} }_{\text{the set of all lower bounds of }A }\Big[b\preceq\text{Inf}(A)\Big]$ - which states that for all lower bounds of $A$, that lower bound "is majorised by" $\text{Inf}(A)$
   • Claim 1: we have part 2 of the definition if and only if $\forall x\in X\Big[\underbrace{\left(\forall a\in A[x\preceq a]\right)}_{x\text{ is a lower bound of }A}\implies x\preceq\text{Inf}(A)\Big]$
   • Claim 2: we claim 1 if and only if $\left(A=\emptyset\vee\Big(\forall x\in X\exists a\in A[x\succ\text{Inf}(A)\implies a\prec x]\Big)\right)$

Notice the $A=\emptyset$ condition here, as in the case $A$ is empty, $\exists a\in A$ is always false. This is a very big caveat.

See also

• Passing to the infimum
• Supremum
  • Passing to the supremum

OLD PAGE

Caution:Rather than trying to fix the old page (which was written with an erroneous claim) I shall instead re-write it and make the caveat known

I got this slightly wrong initially, I was taught that an infimum is the greatest lower bound, that would mean that $\text{Inf}(A)$ was a lower bound such that any value greater than $\text{Inf}(A)$ would fail to be a lower bound (thus $\text{Inf}(A)$ is the greatest one, as any bigger fail to be). This leads to the formulation of $\text{Inf}(A)$ as:

• $\forall x\in X\exists a\in A[x>\text{inf}(A)\implies a<x]$ (If you pick a value greater than the inf, there exists an element in $A$ less than what you picked) and
• $\forall a\in A[\text{inf}(A)\le a]$ (the inf is actually a lower bound)

However there is a problem, the book I was reading speaks about $\text{Inf}(\emptyset)$, if $A:=\emptyset$ then the expression:

• $\exists a\in A$

cannot be true (there does not exist anything in $A$ at all! Let alone something that satisfies the rest of the statement!).

I must make this caveat very clear in the new version

OLD PAGE START

Stub grade: A

This page is a stub

This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:

Needs fleshing out, INCOMPLETE PAGE

A closely related concept is the supremum, which is the smallest upper bound rather than the greatest lower bound.

Definition

An infimum or greatest lower bound (AKA: g.l.b) of a subset $A\subseteq X$ of a poset $(X,\preceq)$:

• $\text{inf}(A)$

such that:

1. $\forall a\in A[\text{inf}(A)\le a]$ (that $\text{inf}(A)$ is a lower bound)
2. $\forall x\in\underbrace{\{y\in X\ \vert\ \forall a\in A[y\le a]\} }_{\text{The set of all lower bounds} }\ \ [\text{inf}(A)\ge x]$ (that $\text{inf}(A)$ is an upper bound of all lower bounds of $A$)
   • Claim 1: , this is the same as $\forall x\in X\exists a\in A[x>\text{inf}(A)\implies a<x]$

Proof of claims

(Unknown grade)

This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.

Please note that this does not mean the content is unreliable. Unless there are any caveats mentioned below the statement comes from a reliable source. As always, Warnings and limitations will be clearly shown and possibly highlighted if very important (see template:Caution et al).
The message provided is:

Make a subpage and put the proof here

See also

• Passing to the infimum
• Supremum
  • Passing to the supremum