Suppose A->D and B->D. Must AB->D?

Question

Let R(ABCD) be a relation...

C. Suppose A⟶D and B⟶D. Must AB⟶D?

D. Suppose AB⟶D. Must A⟶D?

I understand that D is incorrect because AB⟶D does not necessarily mean A⟶D and B⟶D but I'm confused then for C. If we are clarifying that in fact A⟶D and B⟶D holds true, then is it safe to say, AB⟶D?

Answer 1

If A⟶D then AX⟶D. Where X stands for anything. The fact that A⟶D is enough. Having said that, you can conclude that even if X⟶D, AX⟶D is still valid. So your first question is answered with: yes.

If AB⟶D then the minimal requirement to determine D is the pair AB. If that is minimal then A does not fulfill the requirement because it is less than the minimal requirement. That means that A⟶D can not be assumed from AB⟶D. So the answer to your second question is: no.