It looks like I need help wrt left and right adjoins.

When thinking about Puzzle 12, I am coming up with the following (and I think it is wrong):

$$f(x) = x*10$$

and

$$g(x) = x/2$$

So, my 'f' maps any natural number from set \\(A\\) into a set of numbers divisible by 10, my set \\(B\\).

My 'g', takes any element of the set of numbers divisible by 10 (the set \\(B\\) ), and maps into some natural number in set \\(A\\)

In my mind, these form a 'Galois connection'.

Because (1) -- they are monotone maps (bigger input produces bigger output),

and (2), they satisfy the $$ f(a) \le_B b \textrm{ if and only if } a \le_A g(b) $$

That is (as an example): \\(f(5)=50\\) and \\(g(50)=25\\). a = 5, b = 50

However, reading this [Wikipedia page](https://en.wikipedia.org/wiki/Galois_connection), it says that these functions should be inverses of each other (which is not the case in my example).

I have to be missing pieces, appreciate, in advance, help with this.

When thinking about Puzzle 12, I am coming up with the following (and I think it is wrong):

$$f(x) = x*10$$

and

$$g(x) = x/2$$

So, my 'f' maps any natural number from set \\(A\\) into a set of numbers divisible by 10, my set \\(B\\).

My 'g', takes any element of the set of numbers divisible by 10 (the set \\(B\\) ), and maps into some natural number in set \\(A\\)

In my mind, these form a 'Galois connection'.

Because (1) -- they are monotone maps (bigger input produces bigger output),

and (2), they satisfy the $$ f(a) \le_B b \textrm{ if and only if } a \le_A g(b) $$

That is (as an example): \\(f(5)=50\\) and \\(g(50)=25\\). a = 5, b = 50

However, reading this [Wikipedia page](https://en.wikipedia.org/wiki/Galois_connection), it says that these functions should be inverses of each other (which is not the case in my example).

I have to be missing pieces, appreciate, in advance, help with this.