Question

determine whether the following argument is valid or invalid. if it is valid, give a proof. if it is invalid, give an assignment of truth values to the variables that makes the premises true and the conclusion false.

$r \vee p$
$r$
$\overline{\
eg p}$

choose the correct answer below.

a. the argument is valid because it is in the form of modus ponens
b. the argument is invalid. when $r$ is true and $p$ is false, the premises are true and the conclusion is false.
c. the argument is valid because it is in the form of disjunctive syllogism.
d. the argument is invalid. when $r$ is true and $p$ is true, the premises are true and the conclusion is false.

Explanation:

First, analyze the argument: premises are $r \lor p$ and $r$, conclusion is $
eg p$.

• For option A: Modus ponens has form $p

ightarrow q, p \vdash q$, which does not match, so A is wrong.

• For option B: If $r$ is true and $p$ is false, premise $r\lor p$ is true, premise $r$ is true, conclusion $

eg p$ is true (not false), so B is wrong.

• For option C: Disjunctive syllogism has form $r\lor p,

eg r \vdash p$ or $r\lor p,
eg p \vdash r$, which does not match, so C is wrong.

• For option D: If $r$ is true and $p$ is true, premise $r\lor p$ is true, premise $r$ is true, conclusion $

eg p$ is false. This makes all premises true and conclusion false, so the argument is invalid.

Answer:

D. The argument is invalid. When r is true and p is true, the premises are true and the conclusion is false.