Question

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

eg p
ightarrow
eg q\equiv
eg q
ightarrow
eg p$: Looking at the truth - table, when $p = T,q = T$, $
eg p
ightarrow
eg q$ is $T$ and $
eg q
ightarrow
eg p$ is $T$; when $p = T,q = F$, $
eg p
ightarrow
eg q$ is $T$ and $
eg q
ightarrow
eg p$ is $F$ (so this is not an equivalence).

• For $q

ightarrow p\equiv
eg q
ightarrow
eg p$: When $p = T,q = F$, $q
ightarrow p$ is $T$ and $
eg q
ightarrow
eg p$ is $F$ (so this is not an equivalence).

• For $

eg p
ightarrow
eg q\equiv q
ightarrow p$: When $p = T,q = F$, $
eg p
ightarrow
eg q$ is $T$ and $q
ightarrow p$ is $T$; when $p = F,q = T$, $
eg p
ightarrow
eg q$ is $F$ and $q
ightarrow p$ is $T$ (so this is not an equivalence).

• For $p

ightarrow q\equiv
eg q
ightarrow
eg p$: By checking all rows of the truth - table, for all combinations of truth - values of $p$ and $q$, the truth - values of $p
ightarrow q$ and $
eg q
ightarrow
eg p$ are the same.

Answer:

$p
ightarrow q\equiv
eg q
ightarrow
eg p$