$($c$)$ Let A and B be propositional formulae. Let U be a set of propositional

formulae. Are the following statements always correct? Whenever they are,

prove this. When a statement is incorrect, provide a counterexample.

$($i$)$ If A is valid, then so is A $$ B$.$

$($ii$)$ If A $$ B is falsifiable, then $$A is satisfiable.

$($iii$)$ If U is unsatisfiable, then so is U $\backslash$cup $\{$A$\}.$