1. Suppose there are three entity sets E1, E2, and E3, and there are many-to-one relationships from E2 to E1 and from E1 to E3. Prove that there exists a many-to-one relationship from E2 to E3.

2. Give real-life examples of the following E/R situations: (i) one entity set E and a one-to-one relationship R from E to itself, (ii) one entity set E and a many-to-many relationship R running from E to itself, (iii) one entity set E and a many-to-one relationship R running from E to itself.

3. Consider that there are three entity sets E1, E2, and E3 and two relationships, denoted by R1 and R2. R1 is one-to-one from E1 to E2. R2 is one-to-one from E2 to E3. Is the following statement true?

• E2 can be safely removed if it doesn't have any attributes and if it doesn't participate in any other relationships. Explain your answer.

4. Design an ER diagram for the following situation:

• Land masses are either islands or continents. All land masses have a name and an area; the name is the key. Some continents are connected to each other, e.g., Asia is connected to Europe.
• No island is connected to any other island or to a continent.
• Bodies of water are either oceans or straits. A body of water has a name (the key) and an area. Islands may be either located in one ocean (e.g., Hawaii is in the Pacific Ocean) or separated from a continent by a strait (e.g., Honshu is separated from Asia by the "Sea of Japan").
• You should not assume that a strait is adjacent to only one continent or to only one island.