Ah! So

**you weren't claiming to prove anything**, just stating what you have to prove? But you called it your "solution", so I assumed you had defined what function you meant by 1, and proved that it is an identity.

Please give it some thought, and

**tell us what you think the identity element ***might *be! This is your task, not ours. And please ask specific questions so we can help you with your thinking.

You appear now to be asking

**what the operation means**. You gave the definition:

(f⊙g)(x) = f(x)⋅g(x)

This means that the "product" of any two functions f and g is the function f⊙g whose value for any input x is the product of the values of the two functions for that input, f(x) and g(x).

Take an example, with n=2. Then B is the set of all functions from {1, 2} to {0, 1}. We can describe such a function by merely listing the values of f(x) for x=1 and x=2; it can be written as an ordered pair, (f(1), f(2)). For example, we can write the function defined by g(1) = 0 and g(2) = 1 as g = (0,1).

Here is the set B: {(0,0), (0,1), (1,0), (1,1)}. Those are all the possible functions (ordered pairs). One of them will be the identity!

To multiply two functions, we just multiply corresponding elements.

The product of the functions f=(1,0) and g=(1,1) is (1⋅1, 0⋅1) = (1,0). Do you follow that? We have (f⊙g)(1)=f(1)⋅g(1)=1⋅1=1, and (f⊙g)(2)=f(2)⋅g(2)=0⋅1=0. In general, f⊙g = (a,b)⊙(c,d) = (ac,bd).

Now, what specific function g could you multiply by

*any *function f, and the result would be the same function, f? That g will be the identity.

By the way, what I am doing is showing you how to think about a problem like this. You first have to make sure you understand what the notation means, which you can commonly do by "playing" with the concepts, often using a small example like my n=2. This makes it more manageable. Once you get an idea of what the problem is about, you can start thinking about the specific questions that are asked.