not "=", equivalent.

the idea is, we have to have a "witness" in the middle. if a~b, and b~c, then b is a witness to: a~c.

so, in your example, we need elements of {1,2,3} who do double-duty as a 1st coordinate, and 2nd coordinate.

1 doesn't qualify, 1 is never a "first coordinate".

2 doesn't qualify, 2 is never a "second coordinate."

so neither 1 nor 2 can be a witness to transitivity.

that just leaves 3. 3 acts as a witness for the pairs, (2,3) and (3,1), so if R is transitive, (2,1) needs to be in R.

(R "passes through 3" that is to say the relation "transits" or "travels" through the middle).

think of transitivity as being the rule: "you can always cut out the "middle man":

a~b~c means a~c.

(a,b) in R is the same thing as saying a~b.