Consider a relation with seven attributes ABCDEGH. The following dependencies are given:AB -> C, AC -> B, AD -> E, B -> D, BC -> A, E -> GWhat is the key?

To find the key of the relation, we need to find an attribute or a set of attributes that can determine all other attributes. This means that the key attribute(s) is on the left side of a functional dependency and all other attributes can be derived from it.

1. Start with all attributes: ABCDEGH
2. Look at the dependencies and remove attributes that are determined by others:
   • AB -> C: Remove C, we get ABDEGH
   • AC -> B: Remove B, we get ADEGH
   • AD -> E: Remove E, we get ADGH
   • B -> D: D is already removed.
   • BC -> A: A is already in the set.
   • E -> G: Remove G, we get ADH
3. Now, we have ADH left. But none of these attributes appear alone on the left side of the dependencies. So, we need to check combinations.
4. Check combinations:
   • AD is on the left side and determines E. E determines G. So, AD can determine E and G. But it can't determine B and C.
   • AH is not on the left side of any dependency. So, it can't be the key.
   • DH is not on the left side of any dependency. So, it can't be the key.
5. So, none of the combinations of ADH can determine all attributes. This means that our key is larger than this set.
6. If we look at the dependencies again, we can see that B and C together can determine A (BC -> A). And A and D together can determine E (AD -> E). And E can determine G (E -> G). So, BC can determine A, D, E, and G.
7. So, the key of the relation is BC.