Consider the set

$\{$T$,$ F$\}$

Now consider some functions on this set. That is$,$ those whose

domains and co$-$domains $($their inputs and outputs$)$ are from this

set.

f $1$

$1=$

$($

F if T

T if F

f $2$

$1=$

$($

T if $\{$T$,$T$\}$

F otherwise

f $2$

$2=$

$($

F if $\{$F$,$ F$\}$

T otherwise

f $2$

$3=$Consider the set

$\{T,F\}$

Now consider some functions on this set. That is$,$ those whose

domains and co$-$domains $($their inputs and outputs$)$ are from this

set.

$f_1^1=\{\begin{array}{cc}F& if\end{array}$

$TifF$

$f_1^2=\{\begin{array}{ccc}T& if\{T,& T\}\end{array}$

$F$ otherwise

$f_2^2=\{\begin{array}{ccc}F& if\{F,& F\}\end{array}$

$T$ otherwise

$f_3^2=\{\begin{array}{ccc}F& if(\text{:}T,& F\text{:})\end{array}$

$T$ otherwise

Is it possible to recover the truth$-$function $f_3^2$ from the truth$-$functions $f_1^1$

and $f_2^2?$ If so$,$ then show how. If not, then explain why not$/$provide a

counterexample.

$($

F if $$T$,$ F$$

T otherwise