Can sets A, B and C be transitive or does transitivity hold just for individual relations. Also does missing relation prove this false?

Question

Can sets A, B and C be transitive or can we describe only individual relations as being transitive. Also does missing a single relation between two sets mean we can't describe the relation as transitive?

So, the following is transitive where the operator is <= (less than or equals) between A and B and B and C

A = {1,2}  B = {2,3}  C = {5,6}

A <= B
Ra 1 <= 2
Rb 2 <= 2
Rc 1 <= 3
Rd 2 <= 3

B <= C
Re 2 <= 5
Rf 2 <= 6
Rg 3 <= 5
Rh 3 <= 6

because all relations between A <= B and B <= C we can say these sets have transitivity and therefore A <= C

however...

if we change set C to {2,5,6} (I added an extra 2)

additional relations between B <= C are as follows

2 <= 2

// 3 <= 2 does not hold so this is not a relation in regards to the operator <= between 3 and 2

So, could we originally describe the relationship of the sets ARB -> BRC as being transitive but now we cannot?

(apologies, please correct any incorrect terminology, I'm self learning this stuff and I have never studied this stuff before! I may have got some of the fundamentals incorrect too)

Answer 1

You need to define what you mean by one set being less than or equal to another. It seems your definition is that all elements of the first set are less than or equal to all the elements of the second, so $A \le B$ is the same as $\max(A) \le \min(B)$ for finite sets. That is a fine definition of a relation between sets of numbers.

Now we can ask if this relation is transitive across all sets of sets. Your example does not show that it is not transitive across all sets of sets because with the changed $C$ you do not have $B \le C$. In fact it is, because given $A\le B$ and $B \le C$ we can show that $A \le C$ by invoking the definition. We are given $\max (A) \le \min (B)$ and $\max (B) \le \min (C)$. Then $\max (A) \le \min (B)\le \max (B) \le \min (C)$ and we have shown that $A \le C$