(X subset of Y or Y subset of X) is transitive?

Question

so I was trying to figure out how to prove something and I failed so because I think the statement is false. anyways here is the statement :

To prove if the relation is Transitive we need to assume xRy and yRv and prove xRv.
so the assumption means that there are 4 options here :

(1) --> if x⊆y and y⊆v then x⊆v

(2) --> if y⊆x and v⊆y then v⊆x
The main problem of the proof comes here :
(3) --> if x⊆y and v⊆y then we can't prove x⊆v or v⊆x (I.e. x={1},v={2},y={1,2,3})
(4) --> if y⊆x and y⊆v then we can't prove x⊆v or v⊆x (I.e. y={1},v={1,2},x={1,3})

So we can't prove for all the scenarios which means its might be not transitive? and if its not transitive then the relation is not considered as Partially ordered set or Equivalence relation.

Sorry for the long post , Am I right that its not transitive?

Thank you!

Answer 1

No, it is not transitive, and the examples you already have in your question confirm that. For example, in your point 3,$xRy$ and $yRv$ but it is not true that $xRv$.

Answer 2

The proof by a counterexample:

Let

$$ \begin{aligned} X &= \{1\}\\ Y &= \,\;\emptyset\\ Z &= \{2\} \end{aligned} $$

Then

• $X\pmb RY$ (because $Y\subset X$), and
• $Y\pmb RZ\,$ (because $Y\subset Z\;$), but
• not $X\pmb RZ$ (because neither $X\subset Z$, nor $Z\subset X$)).

Thus, the relation $\pmb R$ is not transitive.