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Equivalence of Functional dependencies states that, if the relations of different Functional dependencies sets are given, then we have to find out whether one Functional dependency set is a subset of other given set or both the sets are equal.

功能依賴項的等價性指出，如果給出了不同功能依賴項集的關系，則我們必須找出一個功能依賴項集是其他給定集的子集還是兩個集相等。

To find out whether one Functional dependency set is a subset of other given set or both the sets are equal.

找出一個功能依賴集是另一個給定集的子集還是兩個集相等。

Suppose R1 and R2 are two Functional dependencies sets for a relation R.

假設R1和R2是關系R的兩個功能依賴集。

If all Functional dependencies of R1 can be derived from Functional dependencies present in R2, we can say that R2 is a subset of R1 (R2 ? R1).

如果可以從R2中存在的功能依賴性中得出R1的所有功能依賴性，則可以說R2是R1的子集(R2 = R1) 。

If all Functional dependencies of R2 can be derived from Functional dependencies present in R1, we can say that R1 is a subset of R2 (R1 ? R2).

如果R2的所有功能相關性都可以從R1中存在的功能相關性中得出，則可以說R1是R2的子集(R1?R2) 。

If 1 and 2 both are satisfied, then R1 = R2.

如果1和2都滿足，則R1 = R2 。

情況1)確定是否為R2?R1 (Case 1) Determining Whether R2 ? R1 or not)

Steps are followed to determine whether R2 is a subset of R1 (R2 ? R1) or not,

遵循步驟確定R2是否是R1的子集(R2?R1)，

Step 1)

第1步)

• Take into consideration, the functional dependencies of set R1.

  考慮到集合R1的功能依賴性。

• For every functional dependency P→ Q, find by using the functional dependencies of set R1, the closure of P.

  對于每個功能依賴項P→Q，通過使用集合R1的功能依賴關系來查找P的閉包。

Step 2)

第2步)

• Take into consideration, the functional dependencies of set R2.

  考慮到集合R2的功能依賴性。

• For every functional dependency P→ Q, find by using the functional dependencies of set R2, the closure of P.

  對于每個功能依賴項P→Q，使用集合R2的功能依賴關系來查找P的閉包。

Step 3)

步驟3)

• Compare the results of Step 1 and Step 2.

  比較步驟1和步驟2的結果。

• If the functional dependency of set R2 has determined all those attributes that were determined by the functional dependencies of set R1, then it means R2 ? R1.

  如果集合R2的功能依賴性確定了所有由集合R1的功能依賴性確定的屬性，則意味著R2 = R1。

• Thus, we conclude R2 is a subset of R1 (R2 ? R1) otherwise not.

  因此，我們得出結論：R2是R1的子集(R2 = R1)，否則不是。

情況2)確定是否R1?R2 (Case 2) Determining Whether R1 ? R2 or not)

Steps are followed to determine whether R1 is a subset of R2 (R1 ? R2),

遵循步驟確定R1是否為R2的子集(R1?R2)，

Step 1)

第1步)

• Take into consideration the functional dependencies of set R2.

  考慮集合R2的功能依賴性。

• For every functional dependency P → Q, find by using the functional dependencies of set R2, the closure of P.

  對于每個功能依賴項P→Q，通過使用集合R2的功能依賴關系來查找P的閉包。

Step 2)

第2步)

• Take into consideration the functional dependencies of set R1.

  考慮集合R1的功能依賴性。

• For every functional dependency P → Q, find by using the functional dependencies of set R1, the closure of P.

  對于每個功能依賴項P→Q，通過使用集合R1的功能依賴關系來查找P的閉包。

Step 3)

步驟3)

• Compare the results of Step 1 and Step 2.

  比較步驟1和步驟2的結果。

• If the functional dependency of set R1 has determined all those attributes that were determined by the functional dependencies of set R2, then it means R1 ? R2.

  如果集合R1的功能依賴性確定了所有由集合R2的功能依賴性確定的屬性，則意味著R1 R R2。

• Thus, we conclude that R1 is a subset of R2 (R1 ? R2) otherwise not.

  因此，我們得出結論，R1是R2的子集(R1?R2)，否則不是。

情況3)確定R1和R2是否彼此滿足 (Case 3) Determining Whether Both R1 and R2 satisfy each other or not)

• If R2 is a subset of R1 and R1 is a subset of R2, then both R1 and R2 satisfied each other.

  如果R2是R1的子集，而R1是R2的子集，則R1和R2彼此滿足。

• Thus, if both the above cases satisfied, we conclude that R1 = R2.

  因此，如果以上兩種情況都滿足，我們可以得出R1 = R2。

基于功能依賴項等效性的示例 (Example based on Equivalence of Functional Dependencies)

Q) A relation R (P, Q, U, S, and T) is having two functional dependencies sets R1 and R2, which is shown as

Q)關系R(P，Q，U，S和T)具有兩個功能依賴項集R1和R2，顯示為

Set R1: Set R2:P → C P → QUPQ → U S → PTS → T

Solution...

解...

Case 1) Determining Whether R2 ? R1 or not

情況1)確定是否為R2?R1

Step 1)

第1步)

• (P)+ = {P, Q, U} // closure of left side of P → QU using set R1.

  (P)+ = {P，Q，U} //使用集合R1關閉P→QU的左側。

• (S)+ = {P, Q, U, S, T} // closure of left side of S → PT using set R1.

  (S)+ = {P，Q，U，S，T} //使用集合R1關閉S→PT的左側。

Step 2)

第2步)

• (P)+ = {P, Q, U} // closure of left side of P → QU using set R2.

  (P)+ = {P，Q，U} //使用集合R2關閉P→QU的左側。

• (S)+ = {P, Q, U, S, T} // closure of left side of S → PT using set R2.

  (S)+ = {P，Q，U，S，T} //使用集合R2關閉S→PT的左側。

Step 3)

步驟3)

Comparing the results of Step 1 and Step 2, we find,

比較步驟1和步驟2的結果，我們發現，

• Functional dependencies of set R2 can determine all the attributes which have been determined by the functional dependencies of set R1.

  集合R2的功能依賴性可以確定由集合R1的功能依賴性確定的所有屬性。

• Thus, we conclude R2 is a subset of R1 i.e. R2 ? R1.

  因此，我們得出結論，R2是R1的子集，即R2 = R1。

Case 2) Determining Whether R1 ? R2 or not

情況2)確定是否R1?R2

Step 1)

第1步)

• (P)⁺ = {P, Q, U} // closure of left side of P→ Q using set R2.

  (P) ⁺ = {P，Q，U} //使用集合R2關閉P→Q的左側。

• (PQ)⁺ = {P, Q, U} // closure of left side of PQ → U using set R2.

  (PQ) ⁺ = {P，Q，U} //使用集合R2關閉PQ→U的左側。

• (S)⁺ = {P, Q, U, S, T} // closure of left side of S → PU and S → T using set R2.

  (S) ⁺ = {P，Q，U，S，T} //使用集合R2關閉S→PU和S→T的左側。

Step 2)

第2步)

• (P)⁺ = {P, Q, U} // closure of left side of P→ Q using set R1.

  (P) ⁺ = {P，Q，U} //使用集合R1關閉P→Q的左側。

• (PQ)⁺ = {P, Q, U} // closure of left side of PQ → U using set R1.

  (PQ) ⁺ = {P，Q，U} //使用集合R1關閉PQ→U的左側。

• (S)⁺ = {P, Q, U, S, T} // closure of left side of S → PU and S → T using set R1.

  (S) ⁺ = {P，Q，U，S，T} //使用集合R1關閉S→PU和S→T的左側。

Step 3)

步驟3)

Comparing the results of Step 1 and Step 2, we find,

比較步驟1和步驟2的結果，我們發現，

• Functional dependencies of set R1 can determine all the attributes which have been determined by the functional dependencies of set R2.

  集合R1的功能依賴性可以確定所有由集合R2的功能依賴性確定的屬性。

• Thus, we conclude R1 is a subset of R2 i.e. R1 ? R2.

  因此，我們得出結論：R1是R2的子集，即R1?R2。

Case 3) Determining Whether Both R1 and R2 satisfy each other or not

情況3)確定R1和R2是否彼此滿足

• From Step 1, we conclude R2 ? R1.

  從步驟1，我們得出R2 2 R1。

• From Step 2, we conclude R1 ? R2.

  從步驟2，我們得出R1 1 R2。

• Thus, we conclude that both R1 and R2 satisfied each other i.e. R1 = R2.

  因此，我們得出結論，R1和R2都彼此滿足，即R1 = R2。

翻譯自: https://www.includehelp.com/dbms/equivalence-of-functional-dependencies.aspx

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