Another very basic Discrete Mathematics homework problem. I don't want the answer as much as I want to understand the question:
For each of the following sets of premises, what relevant conclusion(s) can be reached? Explain which rules of inference are used.
a) "If I play hockey, then I am sore the next day", "I use the whirlpool if I am sore", "I did not use the whirlpool"
b) "I am dreaming or hallucinating", "I am not dreaming", "If I am hallucinating, I see elephants smoking"
Okay, now my problem is with b, which ENDS with a conditional. I'm pretty confident that I already got a) correct, so let's look at b):
- $p$: I am dreaming
- $q$: I am hallucinating
- $r$: I see elephants smoking
According to the question, we have:
- $p$ V $q$
- $q\rightarrow r$
The top two premises can be shortened to simply $q$ via "disjunctive syllogism":
- $q \rightarrow r$
So...which rule can you use to draw any conclusions from the above, and what is the conclusion?
Using a truth table, if we look at the row where $q$ AND $q\rightarrow r$ are true, this means that $r$ must be true. So...is the conclusion $r$? But what rule is that?