Скачать или смотреть Transitive Relation

REMARKS:
1. ≤ is transitive, because if x ≤ y and y ≤ z, then x ≤ z.

2. The relation in the example is actually also reflexive and symmetric. You can try verifying the latter two properties yourself as an exercise.

3. I used rather simplistic and contrived examples in the videos on reflexive and symmetric relations because those were easy to verify, but transitive relations are a bit more annoying and unsatisfying to just check by enumerating every case, so I decided to use a more interesting example for this video.

4. Similar to symmetric relations, the empty set is also transitive. The reason is because the conditional statement in the definition of a transitive relation is vacuously true (xRy and yRz is always false, because aRb is always false for any a, b ∈ A).

5. 'Similarity' in geometry is also transitive. If triangle A is similar to triangle B and triangle B is similar to triangle C, then triangle A must be similar to triangle C.

6. The 'has the same birthday as' relation is also transitive. If Harry has the same birthday as Ron and Ron has the same birthday as Hermione, then of course Harry and Hermione also share the same birthday.

7. Equality is also transitive, because if x = y and y = z, then x = z.

0:00 - 1:10 Introduction
1:11 - 5:06 Example