Option C is the only FALSE statement.
We can always have a lossless decomposition into $\textsf{BCNF}$ but not always we can have a lossless and dependency preserving decomposition. But this is always possible in the case of $\textsf{3NF}.$
Option A is true as the requirement of $\textsf{BCNF}$ required a relation schema to be in $\textsf{3NF}.$ Actually $\textsf{3NF}$ allows transitive dependency for prime attributes whereas $\textsf{BCNF}$ does not.
Option D is true as shown below.
Assume the two attributes to be $A$ and $B.$
Now, we can have three cases:
- Either $A$ or $B$ is the candidate key but not both. i.e., $A \to B$ or $B \to A.$ No other $\text{FD}$ is possible and $\text{LHS}$ of all $\text{FDs}$ are superkeys and so $\textsf{BCNF}$ requirement is satisfied.
- Both $A$ and $B$ are candidate keys. i.e., $A \to B$ and $B \to A.$ Like in above case $\textsf{BCNF}$ requirement is satisfied.
- Neither $A \to B$ nor $B \to A$ and so $AB$ is the key. So, no other $\text{FD}$ is possible and this case also satisfies $\textsf{BCNF}$ requirement.
Thus any relation with $2$ attributes is guaranteed to be in $\textsf{BCNF}.$
Ref: https://gatecse.in/demystifying-database-normalization/