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.
$r \
ightarrow \
eg q$
$q$
$\overline{\
eg r}$

choose the correct answer below.
a. the argument is invalid. when r is false and q is false, the premises are true, and the conclusion is false.
b. the argument is valid because it is in the form of modus tollens.
c. the argument is invalid. when r is true and q is true, the premises are true, and the conclusion is false.
d. the argument is valid because it is in the form of disjunctive syllogism.

Explanation:

First, recall the structure of the argument: premises are $r
ightarrow
eg q$ and $q$, conclusion is $
eg r$. Modus tollens follows the form: if $P
ightarrow Q$ and $
eg Q$, then $
eg P$. Here, let $P = r$ and $Q =
eg q$, so $
eg Q = q$. The argument exactly matches modus tollens, so it is valid. We can also verify truth values: for the conclusion $
eg r$ to be false, $r$ must be true. If $r$ is true, for $r
ightarrow
eg q$ to be true, $
eg q$ must be true (so $q$ is false), which contradicts the premise $q$ being true. Thus, no truth assignment makes premises true and conclusion false, confirming validity.

Answer:

B. The argument is valid because it is in the form of modus tollens.