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 and is a lossless join that the join of and on the common attribute A is equal to the original relation R, and that there is no loss of information in the decomposition.Given the functional dependencies in set F: We can see that the attribute A in and is a common attribute, and it determines the attributes B, C, and E in , and the attribute D in . Therefore, when we join and on the attribute A, we will get the original relation R without any loss of information.Thus, the into and is a lossless join decomposition.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.Given the functional dependencies in set F: We can see that the attribute A in and is a common attribute, and it determines the attributes B, C, and E in , and the attribute D in . Therefore, we can decompose R into two relations R_{2}=(A,D,E) R_{1}=(A,B,C) R_{1} R_{2}=(A,D,E) R_{2} R_{1}=(A,B,C) R_{2}=(A,D,E)$.