Chapter B.2, Problem 11E

Solution

(a.)

The truth value of $p$ .

It has been determined that $p$ is false.

Given:

  $p\to q$ , $q\to r$ and $r\to s$ are true, but $s$ is false.

Concept used:

According to Chain rule, if $p\to q$ and $q\to r$ are true, then $p\to r$ is true.

According to Modus Ponens, if $p\to q$ and $p$ are true, then $q$ is true.

According to Modus Tollens, if $p\to q$ is true and if $q$ is false, then $p$ is false.

Calculation:

It is given that $p\to q$ and $q\to r$ are true.

Then, by Chain rule, $p\to r$ is true.

Now, $p\to r$ is true and it is also given that $r\to s$ is true.

Then, by Chain rule, $p\to s$ is true.

Now, $p\to s$ is true and it is given that $s$ is false.

Then, by Modus Tollens, $p$ is false.

Conclusion:

It has been determined that $p$ is false.

(b.)

The truth value of $p$ .

It has been determined that $p$ is false.

Given:

  $\left(p\wedge q\right)\to r$ is true, $r$ is false and $q$ is true.

Concept used:

According to Chain rule, if $p\to q$ and $q\to r$ are true, then $p\to r$ is true.

According to Modus Ponens, if $p\to q$ and $p$ are true, then $q$ is true.

According to Modus Tollens, if $p\to q$ is true and if $q$ is false, then $p$ is false.

Calculation:

It is given that $\left(p\wedge q\right)\to r$ is true and $r$ is false.

Then, by Modus Tollens, $p\wedge q$ is false.

Now, $p\wedge q$ is false and it is given that $q$ is true.

According to the property of $\wedge$ , if $q$ is true, then $p\wedge q$ is false if and only if $p$ is false.

Hence, $p$ is false.

Conclusion:

It has been determined that $p$ is false.

(c.)

If $q$ can be true.

It has been determined that $q$ is true.

Given:

  $p\to q$ is true and $q\to p$ is false.

Concept used:

According to Chain rule, if $p\to q$ and $q\to r$ are true, then $p\to r$ is true.

According to Modus Ponens, if $p\to q$ and $p$ are true, then $q$ is true.

According to Modus Tollens, if $p\to q$ is true and if $q$ is false, then $p$ is false.

Calculation:

It is given that $q\to p$ is false.

This is only possible if $q$ is true and $p$ is false.

This also satisfies the given fact that $p\to q$ is true.

Hence, $q$ is true.

Conclusion:

It has been determined that $q$ is true.