# If AB->C is an FD for a database then does that imply A->C

From below link on Functional Dependencies under FD Axioms(Armstrong's)

I came to know that ab→ c does NOT imply a → c and b → c

But, when I was reading the book "An Introduction to Database Systems" by C.J.Date, on page 279, there is an example that tells about reduction of following set of FDs:

A -> BC
B -> C
A -> B
AB -> C
AC -> D

and under point number 3 of the reduction process it further mentions: Next, we observe that the FD AB -> C can be eliminated, because again we have A -> C, so AB -> CB by augmentation, so AB -> C by decomposition.

So, that means that if A -> C, then we can imply AB -> C.

Assuming we are given the FD AB -> C in a separate case, then let us assume that A -> C, so AB -> C by above deduction, thus our assumption is true. Then, it proves the data in above link false! Is this mathematical approach not correct or I am wrong in a different way? Please throw some light on each perspective!

I came to know that `ab → c` does NOT imply `a → c` and `b → c`

Correct. It does not imply either.

....
So, that means that if A -> C, then we can imply AB -> C.

Yes, correct. but the following:

Assuming ... Is this mathematical approach not correct or I am wrong in a different way?

No, `A -> C` is deducted from `A -> BC`, not from `AB -> C`.

`A -> BC` is just a shorthand notation for `A -> B and A -> C`

• But from above discussion, if A -> C then AB -> C, so if AB -> C holds then A -> C also holds pertaining to the same case, thus, if AB -> C then A -> C? Take it like this also - AB -> C then AB->CB by augmentation, and thus A -> C by Composition rule gone backwards (if A -> B and C -> D then AC -> BD, the same rule going backwards i.e.)? – vinaych Feb 25 '17 at 14:25
• `if A -> C then AB -> C`: Correct. `if AB -> C then A -> C`: Wrong. All men are human but not all human are men. – ypercubeᵀᴹ Feb 25 '17 at 14:27
• In your men,human case which is which ? A is men and C is human I suppose? – vinaych Feb 25 '17 at 14:29
• No, let me rephrase. A=red-hair, B=boy, C=male. Now AB->C (red-hair boy->male) is true. But that doesn't mean that A->C (red-hair -> male) is true. – ypercubeᵀᴹ Feb 25 '17 at 15:55

I came to know from below question:

and concluded where I was wrong, where, I was going the reverse way! AB->C thus, does not imply A->C just by assuming A->C and proving AB->C, in a similar way as above question, where, AX->BX cannot imply A->B)