# What does the free variable t represent in Tuple Relational Calculus?

Consider the query:

Find all students who have taken all courses offered in the Biology department

In the book the query is given as:

$$\{t\ | \ \exists r \in student \ (r[ID] = t[ID]) \ \land \ (\forall u \in course \ (u[dept\_name]\ = \ "Biology" \ \Rightarrow \exists s \in takes(t[ID] = s[ID] \ \land s[course\_id] = u[course\_id]))\}$$

Now I have two questions here :

1) $$t$$ is a free variable and thus any tuple(including those not in the mentioned relations) can be represented using $$t$$. But how is it guaranteed that $$t$$ will only contain the attribute $$ID$$? Are we defining $$t$$ to contain $$ID$$ only and the way to do that is by using $$r[ID] = t[ID]$$ ?

2) Why is it necessary to include $$t[ID] = s[ID]$$? And is the value of this $$t[ID]$$ the same as in $$r[ID] = t[ID]$$?