Prove or disprove that `T=R÷S` implies `S=R÷T`? [closed]
-
29-09-2020 - |
Вопрос
Prove or disprove:
Given that
T=R÷S
, it is implied thatS=R÷T
.
My attempt:
Let us take some R,S
tables and we will divide R÷S
:
step 1:
R S ==> T=R÷S
------------- --------- -----
| A | B | C | | A | B | | C |
| 1 | 2 | 4 | | 1 | 2 | | 4 |
| 3 | 3 | 4 | | 3 | 3 | | 9 |
| 1 | 7 | 4 |
| 1 | 2 | 9 |
| 1 | 2 | 8 |
| 3 | 3 | 5 |
| 4 | 1 | 2 |
| 3 | 3 | 9 |
step 2: lt's check R÷T:
R T ==> S
------------- ----- -------
| A | B | C | | C | | A | B |
| 1 | 2 | 4 | | 4 | | 1 | 2 |
| 3 | 3 | 4 | | 9 | | 3 | 3 |
| 1 | 7 | 4 |
| 1 | 2 | 9 |
| 1 | 2 | 8 |
| 3 | 3 | 5 |
| 4 | 1 | 2 |
| 3 | 3 | 9 |
therefore the answer is YES (according to my attempt)
Is my attempt correct?
Решение
First, your answer is "Yes" but without any proof, only a single example. You have proven nothing about the general case. If I were to mark your answer, you'd get a straight 0 (OK, maybe a 1 because you understand what relational division is and how to perform it) and a note to read up on mathematical proofs.
For the actual question, the answer is: No, it is not correct. Now to prove that, we do only need a single counter-example. Here it is:
R S ==> T=R÷S
------------ --------- -----
| A | B | C | | A | B | | C |
| 1 | 2 | 4 | | 1 | 2 | | 4 |
| 1 | 2 | 9 | | 9 |
| 3 | 3 | 4 |
| 3 | 3 | 8 |
| 3 | 3 | 9 |
| 4 | 1 | 2 |
But:
R T ==> R÷T != S
------------ ----- ---------
| A | B | C | | C | | A | B |
| 1 | 2 | 4 | | 4 | | 1 | 2 |
| 1 | 2 | 9 | | 9 | | 3 | 3 |
| 3 | 3 | 4 |
| 3 | 3 | 8 |
| 3 | 3 | 9 |
| 4 | 1 | 2 |
This is actually similar to integer division (when we discard the remainder). Sometimes it's Yes:
37 / 7 = 5 (the remainder 2 is discarded) and
37 / 5 = 7 (the remainder 2 is discarded)
and sometimes is No:
37 / 10 = 3 (the remainder 7 is discarded) but
37 / 3 = 12 (the remainder 1 is discarded)
and of course 12 != 10
, and the single counter-example disproves the general case.
What can be proven is this:
- If
R÷S = T
, thenS
is a (not necessarily strict) subset ofR÷T
.