Hey guys, I'm kind of stuck here. Any hint will be of great help

Let be a finite set and the boolean algebra of boolean functions of . Im trying to show that there is an identification between and the truth valuation of the boolean functions if X is finite. In this case a truth valuation is a morphism between (all functions from to ) and . To proof this im going to use the function and proof its bijective.

I proved that is one to one, but im having a hard time proving onto. I took a function and then the set . I then wanted to used the fact that the sets are generated by a single element to then construct so that . But realized that since X is not an algebra I cant take the product and also that since the elements of are not morphism, I cant guarantee that is not empty.

Any help?