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) 
Reference Link on research

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 -&gt; C is deducted from A -&gt; BC, not from AB -&gt; C.

A -&gt; BC is just a shorthand notation for A -&gt; B and A -&gt; C