Identify and explain all significant logical errors in the proofs below, listing the line number$($s$)$ where the error is first introduced. If the original claim is true, explain how to correct the proof; otherwise, provide a counterexample showing that the claim is false with brief justification $(2-3$ sentences$).$ By $$significant$$ we mean an error which misapplies rules of inference. Referring to a wrong line number or unusual formatting, for example, are not significant. To show a claim is false, one should provide a case in which all the givens are true, the conclusion evaluates to false. This may involve assigning truth values to propositions and defining predicates $($so that quantified predicate logic statements have a well$-$defined truth value$).$