$5\mathrm{.}9$ Populating Truth Tables with Boolean Functions

IMPORTANT: You will need to use the drop down above the code, below, to select $"$PA$2.$py$"$ and paste your PA$2$ code in that file in order for your PA$3$ code to work.

For this assignment you will be provided with a function callf$2($f$,$p$,$q$)$ whose input is a function and its two arguments, and returns the value of the function applied to the arguments. You are also given a "main" to test your code.

Your task is to implement three functions: eval$\_$tt$\_$f$2($f$)$ and equivalent$($tt$1,$tt$2)$ and is$\_$tautology$($tt$).$

The function eval$\_$tt$\_$f$2($f$)$ will, given a two argument Boolean function f as input $($one of the functions you created in PA$2.$py$),$ return the truth table for f$.$ Use the function make$\_$tt$\_$ins$($n$)$ function you implemented in PA$2.$py to create a list with the inputs for the truth table, and then use callf$2$ to append the truth value for f to each row. For example, evalttf$2($iff$)$ should return:

$[[$False$,$ False, True$],[$False$,$ True, False$],[$True$,$ False, False$],[$True$,$ True, True$]]$

The function equivalent$($tt$1,$tt$2)$ should return True if tt$1$ and tt$2$ are logically equivalent and False otherwise. For example, equivalent$($eval$\_$tt$\_$f$2($PA$2.$implies$),$ eval$\_$tt$\_$f$2($PA$2.$nqIMPnp$))$

should return True.

The function is$\_$tautology$($tt$)$ should return True if the tt parameter has a True value as the output for all possible tt inputs, otherwise it should return false.