Question

use the truth table to determine which statements are equivalent. select all correct options. q→p≡¬q→¬p ¬p→¬q≡q→p ¬p→¬q≡¬q→¬p p→q≡¬q→¬p

Explanation:

Step1: Recall logical - equivalence concept

Two logical statements are equivalent if they have the same truth - values for all possible combinations of truth - values of their variables.

Step2: Analyze each option using the truth - table

For the first option \(q
ightarrow p\) and \(
eg q
ightarrow
eg p\): From the truth - table, when \(p = T\) and \(q = F\), \(q
ightarrow p\) is \(T\) and \(
eg q
ightarrow
eg p\) is \(F\), so they are not equivalent.
For the second option \(
eg p
ightarrow
eg q\) and \(q
ightarrow p\): From the truth - table, when \(p = F\) and \(q = T\), \(
eg p
ightarrow
eg q\) is \(F\) and \(q
ightarrow p\) is \(T\), so they are not equivalent.
For the third option \(
eg p
ightarrow
eg q\) and \(
eg q
ightarrow
eg p\): From the truth - table, when \(p = F\) and \(q = T\), \(
eg p
ightarrow
eg q\) is \(F\) and \(
eg q
ightarrow
eg p\) is \(T\), so they are not equivalent.
For the fourth option \(p
ightarrow q\) and \(
eg q
ightarrow
eg p\): Looking at the truth - table, for all combinations of \(p\) and \(q\) (i.e., \((T,T)\), \((T,F)\), \((F,T)\), \((F,F)\)), the truth - values of \(p
ightarrow q\) and \(
eg q
ightarrow
eg p\) are the same.

Answer:

\(p
ightarrow q\equiv
eg q
ightarrow
eg p\)