Question

use a truth table to determine whether the two statements are equivalent ~a→~b, ~b→a
construct a truth table for ~a→~b

Explanation:

Step1: Recall implication rule

The implication $p
ightarrow q$ is false when $p = T$ and $q = F$, and true otherwise. Also, $\sim T=F$ and $\sim F = T$.

Step2: Fill truth - table for $\sim a

ightarrow\sim b$
When $a = T$ and $b = T$, $\sim a=F$, $\sim b=F$, so $\sim a
ightarrow\sim b$ is $F
ightarrow F=T$.
When $a = T$ and $b = F$, $\sim a=F$, $\sim b=T$, so $\sim a
ightarrow\sim b$ is $F
ightarrow T=T$.
When $a = F$ and $b = T$, $\sim a=T$, $\sim b=F$, so $\sim a
ightarrow\sim b$ is $T
ightarrow F=F$.
When $a = F$ and $b = F$, $\sim a=T$, $\sim b=T$, so $\sim a
ightarrow\sim b$ is $T
ightarrow T=T$.

Step3: Fill truth - table for $\sim b

ightarrow a$
When $a = T$ and $b = T$, $\sim b=F$, so $\sim b
ightarrow a$ is $F
ightarrow T=T$.
When $a = T$ and $b = F$, $\sim b=T$, so $\sim b
ightarrow a$ is $T
ightarrow T=T$.
When $a = F$ and $b = T$, $\sim b=F$, so $\sim b
ightarrow a$ is $F
ightarrow F=T$.
When $a = F$ and $b = F$, $\sim b=T$, so $\sim b
ightarrow a$ is $T
ightarrow F=F$.

Step4: Compare truth - values

Since the truth - values of $\sim a
ightarrow\sim b$ and $\sim b
ightarrow a$ are not the same for all combinations of $a$ and $b$, the two statements are not equivalent.

Answer:

The two statements $\sim a
ightarrow\sim b$ and $\sim b
ightarrow a$ are not equivalent.