(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
According to Modus Ponens, if
According to Modus Tollens, if
Calculation:
Let
It is given that “All college students are poor”.
Then,
It is also given that “Helen is a college student”.
Then, for Helen,
Now, for Helen,
Then, according to Modus Ponens, it follows that
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
According to Modus Ponens, if
According to Modus Tollens, if
Calculation:
Let
It is given that “Some freshmen like mathematics”.
Then,
It is also given that “All people who like mathematics are intelligent”.
Then,
Now, for some freshmen,
Then, according to Modus Ponens, it follows that
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
According to Modus Ponens, if
According to Modus Tollens, if
Calculation:
Let
It is given that “If I study for the final, then I will pass the final”.
This implies that
It is also given that “If I pass the final, then I will pass the course.”
This implies that
Finally, it is given that “If I pass the course, then I will look for a teaching job.”
This implies that
Now,
Then, according to the Chain rule,
Now,
Then, according to the Chain rule,
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
According to Modus Ponens, if
According to Modus Tollens, if
Calculation:
Let
It is given that “Every equilateral triangle is isosceles”.
Then,
It is also given that “There exist triangles that are equilateral”.
Then, for triangles,
Now, for some triangles,
Then, according to Modus Ponens, it follows that
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.
Chapter B.2 Solutions
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