Are these conditional statements true?

Question

This chapter is brewing in me a dislike for my Math book, sort of. It seems the reasoning applied to solve a particular problem is different from that used to solve another one. I've come with my issues to this site twice before (regarding conditional statements, logic, and proofs) and the answers provided by members on this site have shown me that it's not just me.

Anyways, here are some exercises I'd like help with.

I'm asking only about the conditional statement of each one, because I can thread with the converse, contrapositive, and inverse by myself.

1. If an angle measures 167 degrees, then it is obtuse.

   • True, because an obtuse angle, by definition, is an angle whose measure is comprised between 90 and 180 degrees.

2. If you are in a Physics class, then you will always have homework.

   • True, because the original statement has communicated it. There is no "outside logic" that makes it true, just the fact that the sentence says "You are in a Physics class, so you always have homework."

3. If I take my driving test, I will get my driver's license.

   • Following the template of n_2 (regarding "outside logic"), I'm inclined to say true; but my gut disagrees, since you need to actually pass the driving test to earn your license.

Answer 1

I can understand your dislike for the book, because 2 and 3 are inferences, rather than if ... then statements.

But, the if... then statements you write in your post are most likely the ones that the book is looking for.

Answer 2

In order for a conditional statement to be false, it must be impossible for the antecedent to be true and consequent false. Since it is possible to be in a Physics class and have no homework, then the second is false. Admittedly, this is only my interpretation, as both words in the phrase "always have" are somewhat ambiguous. Also, the third statement is false, for precisely the reason that you suspected, and that is non-ambiguous.