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功能依赖 (Functional Dependency)

A relational Database management System (RDBMS) represents the database o a collection of relations/tables. A functional dependency is a constraint between two sets of attributes in a relation. It is the property of semantics or meaning of attribute. Functional dependency is also a property of relational schema "R" and not a particular legal relation "r" of "R".

关系数据库管理系统(RDBMS)代表数据库是关系/表的集合。 功能依赖性是关系中两组属性之间的约束。 它是语义的属性或属性的含义。 功能依赖性也是关系模式“ R”的属性，而不是“ R”的特定法律关系“ r” 。

Now let us consider, a relational "a" schema "R" and let "x" and "y" be the two set of attributes, now there is a functional dependency from "x" to "y"

现在让我们考虑一个关系“ a”模式“ R”，并让“ x”和“ y”成为两组属性，现在存在一个从“ x”到“ y”的功能依赖关系。

If, t1[x] = t2[x] then, t1[y] = t2[y]

如果t1 [x] = t2 [x]，则t1 [y] = t2 [y]

Here, "x" is the determinant and "y" is the dependent or "x" determines "y".

在此 ，“ x”是行列式，“ y”是从属项，“ x”确定“ y”。

• Some functional dependency is directly visible in relational model but some either set of dependencies also hold good which are not directly visible.

  一些功能依赖关系在关系模型中是直接可见的，但是某些依赖关系集也可以保持良好状态，而不能直接看到。

• The entire set of functional dependency is called as complete set.

  整个功能依赖性集称为完整集。

• Therefore, we must know how to calculate closure set of functional dependencies before Normalization.

  因此，我们必须知道如何在归一化之前计算函数依赖的闭包集。

• A functional dependency from "A" to "B" is said to be trivial if "B" is a subset of "A".

  如果“ B”是“ A”的子集，则从“ A”到“ B”的功能依赖关系被认为是微不足道的。

Example: In the corresponding relation,

示例：在对应关系中，

    Tuple   A       C
t1      a1      c1
t2      a1      c1
t3      a2      c2
t3      a2      c2
t4      a3      c3
t5      a3      c2
A → C holds as
t1[A] = t2[A] then t1[C] = t2[C]
t3[A] = t3[A] then t3[C] = t4[C]
but, C → A does not holds as.
t3[C] = t4[C] = t3[C] but t3[A] = t4[A] does not equal t5[A].

如何查找功能依赖项有效还是无效？ (How to find whether a functional Dependency is valid or invalid?)

There are some steps to find whether a functional dependency is valid or invalid:

有一些步骤可以确定功能依赖项是有效的还是无效的 ：

Step 1) Check if all the values of "A" are distinct than it is valid otherwise invalid.

步骤1)检查“ A”的所有值是否均不同，否则有效。

Step 2) Check if all the values of the "B" are same than the functional dependency is also valid.

步骤2)检查“ B”的所有值是否均与功能依赖项相同是否有效。

Step 3) Otherwise we have to find at least one value of "A" on which are having different values of "B" than the functional dependency is invalid.

步骤3)否则，我们必须找到至少一个值“ A”，在该值上具有不同于功能依赖项无效的“ B”值。

属性关闭 (Attribute Closure)

The set "A*" is said to be the closure set of "A" if the set of attributes are functionally dependent on the attributes of "A"

如果属性集在功能上取决于“ A”的属性，则将集合“ A *”称为“ A”的闭包集合。

一些计算封闭集的推理规则 (Some inference rules to calculate the closure set)

1. Reflexive rule: A rule is said to be reflexive if B is a subset of a then A → B. This is called trivial functional dependency rule.

   自反规则：如果B是A→B的子集，则规则说是自反的。这称为琐碎功能依赖规则。

2. Augmentation rule: A rule is said to be augmented if A → B then Ar → Br holds good.

   增强规则：如果A→B然后Ar→Br保持良好，则可以说是增强规则。

3. Transitive rule: A rule is said to be transitive if A → B, B → r then A → r.

   传递规则：如果A→B，B→r然后是A→r，则称为传递规则 。

These three rules are functionally complete and known as RAT rule or RAT axioms and also called Armstrong rule which means only theses rules are sufficient enough to find closure set.

这三个规则在功能上是完整的，称为RAT规则或RAT公理，也称为Armstrong规则，这意味着仅这些规则足以找到闭合集。

4) Union rule

4)联盟规则

A rule is said to be union if A → B, B → r then A → Br also holds good.

如果A→B，B→r那么A→Br也成立，则规则称为并集。

1. Decomposition rule: A rule is said to be decomposed if A → Br, A → B then A → r.

   分解规则：如果A→Br，A→B然后A→r，则称为分解规则 。

2. Pseudo transitivity rule: If A → B and BW → r then WA → r.

   伪传递性规则：如果A→B和BW→r，则WA→r。

翻译自: https://www.includehelp.com/dbms/functional-dependency-and-attribute-closure.aspx

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