R(A B C D E)

F = {A→ BC, CD→ E, E→A}

Which of the following is correct?

The correct option is AD, ED, CD are CK (Candidate Keys).

To determine the Candidate Keys (CK) of the relation $R\left(A,B,C,D,E\right)$ with the functional dependencies (FDs) $F=\left\{A\to BC,CD\to E,E\to A\right\}$, we need to find the attribute closures that can determine all the attributes of the relation $\left\{A,B,C,D,E\right\}$.

Let's analyze the attribute closures for the combinations given in the correct option:

1. Finding the closure of AD, denoted as ${\left(AD\right)}^{+}$:
- Start with the initial set: ${\left(AD\right)}^{+}=\left\{A,D\right\}$.
- Since $A\to BC$, we add $B$ and $C$ to the closure: ${\left(AD\right)}^{+}=\left\{A,D,B,C\right\}$.
- Now we have $C$ and $D$ in the closure. Since $CD\to E$, we can add $E$ to the closure: ${\left(AD\right)}^{+}=\left\{A,D,B,C,E\right\}$.
Since ${\left(AD\right)}^{+}$ contains all the attributes of relation $R$, and no proper subset of $AD$ (i.e., $A$ or $D$ individually) can determine all attributes, AD is a Candidate Key.

2. Finding the closure of ED, denoted as ${\left(ED\right)}^{+}$:
- Start with the initial set: ${\left(ED\right)}^{+}=\left\{E,D\right\}$.
- Since $E\to A$, we add $A$ to the closure: ${\left(ED\right)}^{+}=\left\{E,D,A\right\}$.
- Since $A\to BC$, we add $B$ and $C$ to the closure: ${\left(ED\right)}^{+}=\left\{E,D,A,B,C\right\}$.
Since ${\left(ED\right)}^{+}$ contains all attributes of $R$ and is minimal, ED is a Candidate Key.

3. Finding the closure of CD, denoted as ${\left(CD\right)}^{+}$:
- Start with the initial set: ${\left(CD\right)}^{+}=\left\{C,D\right\}$.
- Since $CD\to E$, we add $E$ to the closure: ${\left(CD\right)}^{+}=\left\{C,D,E\right\}$.
- Since $E\to A$, we add $A$ to the closure: ${\left(CD\right)}^{+}=\left\{C,D,E,A\right\}$.
- Since $A\to BC$, we add $B$ to the closure: ${\left(CD\right)}^{+}=\left\{C,D,E,A,B\right\}$.
Since ${\left(CD\right)}^{+}$ contains all attributes of $R$ and is minimal, CD is a Candidate Key.

Therefore, AD, ED, and CD are all Candidate Keys of the relation R.