Let $S$ be a set, A and $B$ be subsets of $S,$ and ${}_T$ represent the characteristic function of

$T,$

${}_A=\{($:$a,1$:$),($:$b,1$:$),($:$c,0$:$),($:$d,0$:$)\},$

${}_B=\{($:$a,0$:$),($:$b,1$:$),($:$c,0$:$),($:$d,1$:$)\},$ determine ${}_{AB}.$

Let

$f$:$RR,f(x)=x^2-2$

$g$:$RR,g(x)=x+4$

$h$:$RR,h(x)=x^3-1$

$(1)$ Determine $g$@$f,f$@$g$;

$(2)$ Are $g$@$f$ and $f$@$g$ injective, surjective or bijective?

$(3)$ Among the functions of $f,g,h,$ which ones have inverse functions? If so$,$ determine these inverse functions.

$3.$ Let $1,2$ and $0,1$ be both real intervals, prove by definition that $[1,2]$~~$[0,1].$

$4.$ Calculate the following expressions based on the set definition of natural numbers:

$(1)36,25$;

$(2)4-3,3o+1$;

$(3)4,1$;

$(4)14,2^2.$

$5.$ Calculate the cardinality of the following sets.

$(1)A=\{x,y,z\}$

$(2)B=\{x|x=n^2{?}^{{?}^{?}}$ninN$\}$;

$(3)C=\{x|x=n^{109}{?}^{{?}^{?}}$ninN$\}$;

$(4)BC$;

$(5)BC$;

$(6)$ The set of all unit circles with centers on the x$-$axis in the plane.