You are given this relation:

A B C
a1 b1 c5
a1 b2 c6
a2 b1 c7
a3 b2 c6

A possible candidate key is: [ Select ] , and a functional dependency is: [ Select ]

Question

You are given this relation:

A	B	C
a1	b1	c5
a1	b2	c6
a2	b1	c7
a3	b2	c6

A possible candidate key is: [ Select ] , and a functional dependency is: [ Select ]

BlackTom AI · Accepted Answer

We start by restating the problem in our own words and listing what we know from the given relation.
- The relation has attributes A, B, and C with the following rows: (a1, b1, c5), (a1, b2, c6), (a2, b1, c7), (a3, b2, c6).
- We need to determine a possible candidate key (a minimal set of attributes that functionally determines all attributes) and a functional dependency of the form (some attributes) --> (other attributes).

First, evaluate whether A alone could be a candidate key. If A were a key, each distinct A value would have exactly one associated B and one C. Looking at the data:
- A = a1 occurs with B = b1 and also with B = b2, giving two different B values for the same A. This means A does not functionally determine B (and thus not C) by itself. Therefore, A alone cannot be a candidate key.

Next, evaluate whether B alone could be a candidate key. If B were a key, each distinct B value would have a unique A and C. Observing the data:
- B = b1 occurs with A = a1 and A = a2, i.e., two different A values for the same B. Hence B does not functionally determine A (and thus not C) by itself. Therefore, B alone cannot be a candidate key.

Now consider the combination AB. If AB is a key, the pair (A, B) should uniquely identify C (and in fact all attributes). Check the four rows:
- (a1, b1) appears once, yielding C = c5.
- (a1, b2) appears once, yielding C = c6.
- (a2, b1) appears once, yielding C = c7.
- (a3, b2) appears once, yielding C = c6.
Since every (A, B) pair is unique and each pair determines C, AB functionally determines all attributes. Therefore AB is a candidate key (and in this dataset, it appears to be the only candidate key).

With a candidate key identified, we can consider a functional dependency that naturally follows from the key. If AB is a key, it functionally determines C as well as A and B (the key’s own components), but most commonly we express the dependency in the form AB --> C to denote that the key uniquely determines the non-key attribute C in this relation. Looking at the rows, whenever the pair (A, B) is the same, the C value must be the same, which confirms AB --> C holds for this dataset.

In summary:
- A alone does not determine B or C, as A is associated with multiple B values (a1 -> b1 and b2).
- B alone does not determine A or C for the same reason (b1 -> a1 and a2).
- The combination AB uniquely identifies each row, so AB is a candidate key.
- Consequently, AB functionally determines C, i.e., AB --> C, which is a natural dependency following from the key.

INFO20003_2025_SM2 Practice Exam Quiz

You are given this relation: A B C a1 b1 c5 a1 b2 c6 a2 b1 c7 a3 b2 c6 A possible candidate key is: [ Select ] , and a functional dependency is: [ Select ]

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