6
$\begingroup$

I was recently introduced to partial order relations. Although I understand the concept of a relation, I do not understand how subsets can have a sequence, or what that has to do with the phrase "ordering of sets by inclusion." Can anyone help me clarify the concept with an intuitive example?

$\endgroup$
4
  • $\begingroup$ It means containment. Llike, $A <B$ if $A \subset B$ $\endgroup$ May 30, 2015 at 7:46
  • $\begingroup$ @Bhaskar Vashishth: Can you elaborate, please? $\endgroup$
    – user142971
    May 30, 2015 at 7:48
  • $\begingroup$ Can you provide a quote of where this phrase appears? I'm not sure what you mean by "ordering of sequence of subset". $\endgroup$
    – augurar
    May 30, 2015 at 7:49
  • $\begingroup$ The story is long @augurar; I'm also confused whether to include the word 'sequence'. So, I gave ?? $\endgroup$
    – user142971
    May 30, 2015 at 7:51

3 Answers 3

7
$\begingroup$

Let $\mathcal{S}$ be a collection of sets. Then for sets $S_1, S_2 \in \mathcal{S}$, we can define $S_1 \prec S_2$ to mean $S_1 \subset S_2$.

So for example, considering sets of integers, we can say $\{1, 2, 3\} \prec \{1, 2, 3, 4\}$. This is a partial order because not all sets are comparable. For instance $\{1, 2, 3\}$ and $\{1, 3, 5\}$ are not comparable because neither is a subset of the other.

$\endgroup$
8
  • $\begingroup$ Thanks, sir. +1 for this. At the first sight, it nailed the doubt. $\endgroup$
    – user142971
    May 30, 2015 at 7:56
  • $\begingroup$ Sir, one question; by ordering what do we get? The definition of set in naive set theory is that it is an unordered collection of mathematical objects. So, what does this "ordering' all mean? $\endgroup$
    – user142971
    May 30, 2015 at 8:17
  • $\begingroup$ @user36790 It's a way of giving the set additional structure. A set along with a partial order is called a partially ordered set. We can then study the properties of this new object. $\endgroup$
    – augurar
    May 30, 2015 at 8:26
  • $\begingroup$ A set along with a partial order means? Suppose $\{a,b,c,d\}$ where $a < b< c$. Then the set along with partial order relation is meaning $\{(a,b),(a,c),(b,c)\}$ this? I am new, so I face problems with these phrases like set along with a relation:| $\endgroup$
    – user142971
    May 30, 2015 at 8:40
  • $\begingroup$ @user36790 It just means we think about the set and the relation together as a single mathematical object. In your comment you provide a set $S$ and a relation $R$ on the set. If we consider these together, we can think of the pair as representing a partially ordered set. $\endgroup$
    – augurar
    May 30, 2015 at 9:26
1
$\begingroup$

Suppose you have a set $\mathtt{S}$. Now denote power set of $\mathtt{S}$ by $\mathcal{P}(\mathtt{S})$. Then $\mathcal{P}(\mathtt{S})$ is a poset under the relation $\subseteq$ (inclusion).

  1. Reflexive- As $A \subseteq A$ for all $A \in \mathcal{P}(\mathtt{S})$
  2. AntiSymmetric- As $A\subseteq B$ and $B \subseteq A$ $\implies A=B$
  3. Transitive- You can see it now, I am sure.

So it is a partial order on $\mathcal{P}(\mathtt{S})$.

On any set of sets, inclusion is a partial order.

$\endgroup$
1
  • $\begingroup$ If set $S$ has less than 3 elements (or even less than 2 elements), can we infer that inclusion is a partial order? $\endgroup$
    – Snowball
    Oct 16, 2020 at 23:04
1
$\begingroup$

Given any family of sets $\mathcal{F}$ there is a poset $P=(\mathcal{F},\{(A,B)\in\mathcal{F}^2:A\subseteq B\})$ corresponding to that family ordered by inclusion. Now by an "inclusion maximal/maximum/minimal/minimum" set in $\mathcal{F}$ what is meant is simply a maximal/maximim/minimal/minimum element of $\mathcal{F}$. Also its worth noting that every partial order is isomorphic to a family of sets ordered by inclusion, in particular the principal lower sets of any partial order when ordered by set inclusion are always isomorphic to said partial order.

$\endgroup$

You must log in to answer this question.