## How to prove that two sets of functional dependencies are equal or not - Proving equality of two sets of functional dependencies - How to find whether two sets of functional dependencies are equal or not? - Equivalent sets of functional dependencies

**Question:**
4. Let F1 = {A → C, AC → D, E → AD} and F2 = {A → CD, E → AH}. Are F1 and F2 are
equivalent?

__Solution:__Equivalent sets of functional dependencies:
Two sets of FDs F
and G over schema R are equivalent, if F
^{+} = G^{+}. That
is, if every functional dependency of F is in G+ and every functional dependency
of G is in F+, then we would say that the sets of functional dependencies F
and G are equivalent. (here, F^{+} and G^{+} means ) - the closure of sets of functional dependenciesFurther, refer here. |

__We first check if F1__

__⊆__

__F2__^{+}:
If we are able to find the closure for
all the determinants of functional dependencies of F2, and if the derived closures
are able to show the functional dependencies of F1, then we say that F1 ⊆
F2

^{+}.
A

^{+}_{F2}= ACD.
That is, if you know

**A**then you know**ACD**according to the FDs of F2. Hence, the functional dependency A → C of F1 is in F2.
AC

^{+}_{F2}= ACD.
That is, if you know

**AC**then you know**ACD**according to the FDs of F2. Hence, the functional dependency AC → D of F1 is in F2.
E

^{+}_{F2}= ACDEH. (from E → AH, A → CD)
That is, if you know

**E**then you know**ACDEH**according to the FDs of F2. Hence, the functional dependency E → AD of F1 is in F2.
From this,

**we are able to derive all the FDs of F1 from F2+. Hence, F1****⊆****F2**is^{+ }**TRUE**.

__We first check if F2__

__⊆__

__F1__^{+}:
We need to follow the above steps for
this too;

A+F1 = ACD.

So, A → CD of F2 is in F1.

E+F1 = ACDE.

So, E → AH of F2 is not in F1.

We are not able to derive all the FDs
of F2 from F. Hence,

**F2****⊆****F1**^{+}**is FALSE**.
From the above, it is proved that

**.**__F1 and F2 are not equivalent__
********************

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