Chapter B.2, Problem 13E

Solution

(a.)

A conclusion that follows logically from the given statement.

It has been determined that a conclusion that follows logically from the given statement is:

Helen is poor.

Given:

All college students are poor.

Helen is a college student.

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:

Let $p$ denote “being a college student” and $q$ denote “being poor”.

It is given that “All college students are poor”.

Then, $p\to q$ is true.

It is also given that “Helen is a college student”.

Then, for Helen, $p$ is true.

Now, for Helen, $p\to q$ and $p$ are both true.

Then, according to Modus Ponens, it follows that $q$ is true for Helen.

Thus, it logically follows that “Helen is poor”.

Conclusion:

It has been determined that a conclusion that follows logically from the given statement is:

Helen is poor.

(b.)

A conclusion that follows logically from the given statement.

It has been determined that a conclusion that follows logically from the given statement is:

Some freshmen are intelligent.

Given:

Some freshmen like mathematics.

All people who like mathematics are intelligent.

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:

Let $p$ denote “liking mathematics” and $q$ denote “being intelligent”.

It is given that “Some freshmen like mathematics”.

Then, $p$ is true for some freshmen.

It is also given that “All people who like mathematics are intelligent”.

Then, $p\to q$ is true for all people including the freshmen referred to above.

Now, for some freshmen, $p\to q$ and $p$ are both true.

Then, according to Modus Ponens, it follows that $q$ is true for some freshmen.

Thus, it logically follows that “Some freshmen are intelligent”.

Conclusion:

It has been determined that a conclusion that follows logically from the given statement is:

Some freshmen are intelligent.

(c.)

A conclusion that follows logically from the given statement.

It has been determined that a conclusion that follows logically from the given statement is:

If I study for the final, then I will look for a teaching job.

Given:

If I study for the final, then I will pass the final.

If I pass the final, then I will pass the course.

If I pass the course, then I will look for a teaching job.

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:

Let $p$ denote “I will study for the final”, $q$ denote “I will pass the final”, $r$ denote “I will pass the course” and $s$ denote “I will look for a teaching job”.

It is given that “If I study for the final, then I will pass the final”.

This implies that $p\to q$ .

It is also given that “If I pass the final, then I will pass the course.”

This implies that $q\to r$ .

Finally, it is given that “If I pass the course, then I will look for a teaching job.”

This implies that $r\to s$ .

Now, $p\to q$ and $q\to r$ are true.

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

Now, $p\to r$ and $r\to s$ are true.

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

Thus, it logically follows that “If I study for the final, then I will look for a teaching job.”

Conclusion:

It has been determined that a conclusion that follows logically from the given statement is:

If I study for the final, then I will look for a teaching job.

(d.)

A conclusion that follows logically from the given statement.

It has been determined that a conclusion that follows logically from the given statement is:

There exist triangles that are isosceles.

Given:

Every equilateral triangle is isosceles.

There exist triangles that are equilateral.

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:

Let $p$ denote “being equilateral triangle” and $q$ denote “being isosceles triangle”.

It is given that “Every equilateral triangle is isosceles”.

Then, $p\to q$ is true.

It is also given that “There exist triangles that are equilateral”.

Then, for triangles, $p$ is true.

Now, for some triangles, $p\to q$ and $p$ are both true.

Then, according to Modus Ponens, it follows that $q$ is true for some triangles.

Thus, it logically follows that “There exist triangles that are isosceles”.

Conclusion:

It has been determined that a conclusion that follows logically from the given statement is:

There exist triangles that are isosceles.