(This is not my question yet.)

Suppose . A set C is said to be closed under if Now suppose . The closure of under is the smallest set such that and is closed under , if there is such a smallest set.

Let

Then the closure of B under f is given by the set

.

Here is my question.

Suppose is a family of functions from A to A, and . The closure of Bunderis the smallest set C such that and C is closed under , if there is such a smallest set.

What should be the closure of B under in this case?

I try to prove, but am unsure.

Should it be or ??

Or could it be ?

I tried to prove is the closure of B under . Every property, closedness under f, and is the smallest set has been proved, except . (I'm kinda stucked.)

I tried to use info from the first part. Let be the closure of B under Then I claimed that is the closure of B under , but my friend said it is not closed. For example,

,

{0,2,4,6,8,10,...}

{0,3,6,9,12,15,...}

does not contain 5, but

Thanks a lot!