also i don't understand this problem. i need help and to prove....the symbol ⊂ is subset.

here the problem:

Prove that There exist an open interval B = (c,d), and sets A and C, with B �⊂ C, and a function f: A-->C such that f^(-1)(B) is not an open interval.

To prove the statement, "There exist an open interval B = (c,d), and sets A and C, with B ⊂ C, and a function f: A-->C such that f^(-1)(B) is not an open interval," you need to follow a step-by-step approach.

Step 1: Understand the problem

- The symbol "⊂" signifies a subset relationship, where B is a subset of C.
- You need to find a specific open interval B, sets A and C, and a function f: A-->C that satisfies the given conditions.
- The goal is to show that the inverse image of B under f, denoted as f^(-1)(B), is not an open interval.

Step 2: Choose an open interval B

- Start by selecting any open interval as B. Let's assume B = (c, d), where c and d are real numbers.

Step 3: Define sets A and C

- Now, define sets A and C, ensuring that B ⊂ C.

Step 4: Define a function f: A-->C

- Define a function f: A-->C such that it satisfies the given conditions. The function should map elements from set A to set C.

Step 5: Find the inverse image of B, f^(-1)(B)

- To find the inverse image of B under f, you need to determine all the elements in A that map to elements in B. This is done by finding f^(-1)(B).
- If f^(-1)(B) is not an open interval, your proof will be complete.

Step 6: Provide the proof

- Clearly state the open interval B, sets A and C, and the function f: A-->C that satisfies B ⊂ C.
- Show the inverse image f^(-1)(B) and explain why it is not an open interval.
- This may involve demonstrating that f^(-1)(B) contains endpoints, or by applying a mathematical property that helps identify when an interval is not open.

By following these steps, you will be able to prove the statement "There exist an open interval B = (c,d), and sets A and C, with B ⊂ C, and a function f: A-->C such that f^(-1)(B) is not an open interval."