Converting between relational calculus and relational algebra

Question

So I'm trying to get my head around relational algebra and calculus and converting between them. I understand that... $$\{ x\mid\exists y.F(x,y) \wedge G(x,y)\}$$ ...equates to $\pi_A F \cap G$

But what about when one of the relations has only a single attribute? Does this affect anything? i.e. $$\{ x\mid\exists y.F(x,y) \wedge G(y)\}.$$

Answer 1

No, I really think it's different.

For instance, let's assume we are working in a bank and we have two relationships Account(Name,balance) and Client(Name,balance)

Let's assume that the balance of a client is unique: it is the sum of the balance of its accounts.

$$\{Name| \exists Balance \ Account(Name,Balance)\wedge Client(Name,Balance) \}$$ Would Give the name of every clients that have a balance equal to the exact sum of one of their account.

Whereas

$$\{Name| \exists Balance \ Account(Balance)\wedge Client(Name,Balance) \}$$

Would give the name of every client that have a balance equal to the exact amount of one of any account. (Am I mispelling?)