Hmmm

Tutorial was interesting this week. The first interesting thing was question 2. The statement is "every course has a prerequisite." Someone said ∃x ϵ C, ∀ y ϵ C, P(x, y). Someone said that's wrong, and it should be the other way around (∀ y ϵ C, ∃x ϵ C, P(x, y)), and most people agreed. Then someone said that it's still true, and people said "wait, it does work." Because the first expression means that there exists one course that is a prerequisite for every course, and that means that every course has a prerequisite. But the two are not equivalent. x ϵ C, ∀ y ϵ C, P(x, y) ⇒ every course has a prerequisite, but every course has a prerequisite does not imply that there exists a course that is a prerequisite of every course. The second interesting thing was the last question in part 1. "Some courses have the same prerequisites." The first answer was ∃x, y, z ϵ C, x ≠ y, P(z, x) ∧ P(z, y). But this only means that some courses have a common prerequisite. ∃x, y ϵ C, x ≠ y, ∀ z ϵ C, P(z, x) ⇔ P(z, y) means some courses have the exact same prerequisites. The third interesting thing was that someone said in English, "only if" is the same as "if and only if". People said not to use English meaning in logic, but he continued. The English meaning is actually correct too. He said if I say "I will quit only if you report me to the media." He said that if I quit, you must have reported me to the media (P ⇒ Q). It's easy to think that if you report me to the media, I will quit (Q ⇒ P), because being reported to the media is scary. But that doesn't have to be true. But he insisted that was the case. Eventually someone said there will be rainbows only if it rains. There can't be rainbows if it doesn't rain, but if it rains, there don't have to be rainbows. Class discussion in tutorials brings up interesting things.