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I took some notes in class that I have trouble understanding. There's a chance my notes are incorrect:

  • The union of a family of sets is the supremum.
  • The intersection is the infimum.

The teacher was talking to us about complete lattices.

I don't understand what a family of sets is supposed to be (maybe the set of all subsets), and I don't understand how the union or the intersection (which result in sets) gives the supremum or the infimum (which are elements). Also, whose supremum and whose infimum?

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2 Answers 2

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Infimum and supremum are just terms for elements in a partially ordered set. There are partially ordered sets whose elements are not numbers, but rather sets. In fact, the real numbers can be represented as sets for themselves.

As in my answer to your previous question, we simply define infimum to be the maximal element which is a lower bound - if it exists.

In the case where sets are used, it is usual to use the $\subseteq$ partial ordering. A family of sets is just a set of sets, some may consider it as a function from an index set $I$ into the partial order, $A_i$ for $i\in I$.

In this case, the supremum is $\bigcup\{A_i\mid i\in I\}=\{x\mid\exists i\in I:x\in A_i\}$. This set is the smallest set which contains all the $A_i$'s.

The same can be done with the intersection, $\bigcap\{A_i\mid i\in I\}=\{x\mid\forall i\in I: x\in A_i\}$.

This union (intersection), if exists in the partial order forms the smallest upper bound (largest lower bound) of the family. It may not exist, in which case the infimum may not exist altogether. However, when considering "all possible sets" with the partial order of inclusion the union and intersection is in fact the supremum/infimum.


Exercise I: Consider $\mathcal P(\mathbb N)$ ordered by $\subseteq$, that is all subsets of the natural numbers.

Now take the family $\{A_i\mid i\in I\}$ defined as $A_i=\{1,i\}$ for $i\in\mathbb N$. The supremum of this family is exactly $\mathbb N$. Can you see why?

Can you calculate the infimum?

Example II: Consider the set of all finite subsets of $\mathbb N$, again ordered by inclusion.

What is the supremum of $\{\{n\}\mid n\in\mathbb N\}$? (that is the set of all singletons) - it has no supremum, since the union of all these sets is an infinite set, which is not a member of the partial order.

Example III: Take the following subsets of $\{1,2,3\}$: $$\big\lbrace\varnothing,\{1\},\{2\},\{1,2,3\}\big\rbrace$$ Order them by inclusion.

What is the supremum of $\big\lbrace\{1\},\{2\}\big\rbrace$? In this case, it is not $\{1,2\}$ since this is not a member of our partial order. It is in fact $\{1,2,3\}$.

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Exercise I: Yes, I can see why, and the intersection is $\{1\}.$ :) Clarification: Whose subset of $\mathbb N$ are these the supremum and infimum of? –  Paul Manta Oct 9 '11 at 19:54
    
@Paul: I'm not sure what sort of clarification you are asking for. –  Asaf Karagila Oct 9 '11 at 19:58
    
Ah nevermind. As I was trying to explain my confusion I understood what I got wrong. :) Thank you for the very good answer and the examples. –  Paul Manta Oct 9 '11 at 20:14
    
Actually, there's one last thing I'd like to make sure. We can talk about suprema not only for individual elements, but also for families of elements; for example, in the set $\{0, 1, 2\},$ the supremum of the family $\{0, 1\}$ is $1$. Right? –  Paul Manta Oct 9 '11 at 20:49
2  
@Paul: Supremum and infimum are always tied into some ordering. Had I wrote $\{a,b,c\}$ instead of $\{1,2,3\}$ - could we deduce the "correct" order of this set? If you order the set as $\{0<1<2\}$, then $1$ is indeed the supremum of $\{0,1\}$. This corresponds to the definition of a supremum. The term "family" is usually reserved to a set of sets in contexts where the underlying elements are not necessarily sets (e.g. numbers). –  Asaf Karagila Oct 9 '11 at 20:53

Supremums and infimums of families of subsets of a given set are based on the order relation $\subseteq$.

For example, $\sup_i B_i$ is the smallest subset $B$ (if this exists, but it does) such that $B$ is larger than $B_i$ for every $i$, that is, such that $B_i\subseteq B$ for every $i$. Of course this is nothing but $B=\bigcup\limits_iB_i$. Likewise for the intersection as the largest subset smaller than every $B_i$.

A difference with the order relation you know on $\mathbb R$ is that two subsets $B'$ and $B''$ are not always comparable in the sense that $B'\subseteq B''$ and $B''\subseteq B'$ may both be false.

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