Homework Assignment Two

For this homework assignment, consider the schema $R=(A,B,C,D,E)$ and the following set

$F$ of functional dependencies holds on $R$ :

$ABC$

$CDE$

$BD$

$EA$

Problem $1[10$ points$].$ Suppose that we decompose the schema $R$ into $R_1=(A,B,C)$ and

$R_2=(A,D,E).$ Show that this decomposition is a lossless$-$join decomposition with

respect to $F.$

Problem $2[10$ points$].$ Suppose that we decompose the relation schema $R$ into

$R_3=(A,B,C)$ and $R_4=(C,D,E).$ Show that this decomposition is not a lossless$-$join

decomposition. Hint: Give an example of a relation $r$ on schema $R$ such that

Problem points$].$ Compute $(BC{)}^{*}.$

Problem $4[20$ points$].$ Compute the canonical cover $F_c.$

Problem $5[10$ points$].$ Show that the decomposition of $R$ into $R_1=(A,B,C)$ and

$R_2=(A,D,E)$ is not a dependency$-$preserving decomposition.

Problem $6[20$ points$].$ Give a lossless$-$join decomposition into BCNF of $R.$

Problem $7[20$ points$].$ Give a lossless$-$join, dependency preserving decomposition

into $3$ NF of $R.$