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We know that every Partial Order Set has to satisfy three conditions :

1- Reflexive

2- Anti-Symmetric

3- Transitive

For example : $$S= \left(\left\{1 \right\},\left\{2 \right\},\left\{3 \right\},\left\{1,2 \right\} \right)$$

And the relation is $\subseteq$ ,My question is why this Set S is a Partial Order Set Althought I Cannot prove that is Transitive which is defined as:

$$R=\{(x,y)\} ; xRy ~\text{and}~ yRz \implies xRz$$ I could not find like : if $\left\{1 \right\} \subseteq \left\{1,2 \right\}\text{ } and \text{ } \left\{1,2 \right\} \subseteq \text{ }\text{ } ?? \text{ }then\text{ } ??$

in another way how can we apply the Transitive test for this set S?

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  • $\begingroup$ Recall the definition of a transitive relation: A relation $R$ on a set $A$ is transitive if whenever $(a,b)\in R$ and $(b,c)\in R \implies (a,c)\in R$ for $a,b,c \in A$. $\endgroup$ – Rohan Dec 8 '17 at 6:54
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Note that statement of the form of "if $A$ then $B$" is true if $A$ is false.

Well, actually we have $\color{blue}{\{1 \}} \subseteq \color{green}{\{ 1,2\}}$ and $\color{green}{\{1,2\}} \subseteq \color{purple}{\{ 1, 2\}}$, then $\color{blue}{\{1\}} \subseteq \color{purple}{\{1,2\}}.$

Notice that for the subset operation, we always have

$$x \subseteq y \text{ and }y \subseteq z \text{ ,then }x \subseteq z$$

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  • $\begingroup$ like that .. i understand the first line but do you mean that y may equel to z? $\endgroup$ – Yaser Tarek Dec 8 '17 at 6:57
  • $\begingroup$ There is no restriction that $x$, $y$ and $z$ have to be distinct right? we are suppose to check all possibilities. though the case for identical elements are trivial. Most important thing should be the last line to show transitivity. $\endgroup$ – Siong Thye Goh Dec 8 '17 at 7:01
  • $\begingroup$ Perfect then i can understand that A relation on a set A: is always Transitive .. right ? Where A is a Set of at least One Element . $\endgroup$ – Yaser Tarek Dec 8 '17 at 7:07
  • $\begingroup$ yes, subset is a transitive relation. $\endgroup$ – Siong Thye Goh Dec 8 '17 at 7:10
  • $\begingroup$ thanks a lot bro.. in the definition of a transitive relation: A relation R on a set A is transitive if whenever (a,b)∈R(a,b)∈R and (b,c)∈R⟹(a,c)∈R(b,c)∈R⟹(a,c)∈R for a,b,c∈A ... could a,b and c be equel ? @rohan $\endgroup$ – Yaser Tarek Dec 8 '17 at 9:43

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