$5.$ Answer the following questions. Justify your answers with a truth assignment $($or two$)$

or an entire table as appropriate.

Note $1$: In this class, when your justification requires a single truth assignment, you

must explicitly give the truth assignment. Don$$t just circle a row in the table. We

want to make sure you know how to pull the appropriate assignment from the table.

Note $2$: If an answer can be justfied with a single truth assignment, you do not need

to provide an entire truth table. You can provide a table if that$$s part of the work you

used to find the assignment, but once you find the assignment, you do not need to finish

the table.

$($a$)$ Is A $->$ B satisfiable?

$($b$)$ Is $($A $$ A$)($B $$ B$)$ satisfiable?

$($c$)$ Is X $->$ X a contradiction?

$($d$)$ Is $$ A $->($A $$ B$)$ a tautology?

$($e$)$ Is $$ A $->($A $$ B$)$ a contingency?

$($f$)$ Is

$($A $->$ B$)($C $$ B$)$

$->$

A $->$ C

a tautology?

$($g$)$ Is

$($A $->$ B$)($C $$ B$)$

$->$

A $->$ C

satisfiable?

$($h$)$ Is

$($A $->$ B$)($C $$ B$)$

$->$

A $->$ C

a contingency?

$($i$)$ Is $($A $->$ B$)->($ A $->$ B$)$ a contradiction?

$($j$)$ Is $($A $->$ B$)->($ A $->$ B$)$ a tautology?

$($k$)$ Is $$ A $$

$($D $$ D$)->$

$($B $$ B$)($C $->$ C$)$

satisfiable?

Hint: No$,$ we don$$t expect you to create an entire $32-$row