3. Suppose we decompose R given in question 1 into R_(1)=(A,B,C) and R_(2)=(A,D,E) show that this decomposition is lossless join decomposition if the functional dependencies in set F given in question 1 hold. F= A-gt BC,CD-gt E,B-gt D,E-gt A 4. Give a lossless join decomposition into BCNF of schema R under given in question 1. F= Aarrow BC,CDarrow gt ,,arrow arrow D,Earrow A

3. To show that the decomposition of R into $R_{1}=(A,B,C)$ and $R_{2}=(A,D,E)$ is a lossless join that the join of $R_{1}$ and $R_{2}$ on the common attribute A is equal to the original relation R, and that there is no loss of information in the decomposition.<br /><br />Given the functional dependencies in set F:<br />$F=\{ A\rightarrow BC,CD\rightarrow E,B\rightarrow D,E\rightarrow A\} $<br /><br />We can see that the attribute A in $R_{1}$ and $R_{2}$ is a common attribute, and it determines the attributes B, C, and E in $R_{1}$, and the attribute D in $R_{2}$. Therefore, when we join $R_{1}$ and $R_{2}$ on the attribute A, we will get the original relation R without any loss of information.<br /><br />Thus, the into $R_{1}$ and $R_{2}$ is a lossless join decomposition.<br /><br />4. To give a lossless join decomposition of schema R into BCNF, we need to decompose R into relations where each relation is in BCNF, and the join of these relations is equal to the original relation R.<br /><br />Given the functional dependencies in set F:<br />$F=\{ A\rightarrow BC,CD\rightarrow E,B\rightarrow D,E\rightarrow A\} $<br /><br />We can see that the attribute A in $R_{1}$ and $R_{2}$ is a common attribute, and it determines the attributes B, C, and E in $R_{1}$, and the attribute D in $R_{2}$. Therefore, we can decompose R into two relations $R_{1}=(A,B,C $R_{2}=(A,D,E)$.<br /><br />Now, let's check if these relations are in BCNF:<br /><br />For $R_{1}=(A,B,C)$:<br />- The attribute A determines the attribute B, so $R_{1}$ is in BCNF.<br /><br />For $R_{2}=(A,D,E)$:<br />- The attribute A determines the attribute D, so $R_{2}$ is in BCNF.<br /><br />Therefore, the lossless join decomposition of schema R into BCNF is:<br />$R_{1}=(A,B,C)$ and $R_{2}=(A,D,E)$.